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MATHEMATICAL  MONOGRAPHS. 

EDITED   BY 

Mansfield  Merriman  and  Robert  S.  Woodward. 

Octavo,  Cloth,  $1.00  each. 

No.  1.    HISTORY  OF  MODERN  MATHEMATICS. 

By  David  Eugene  Smith. 

No.  2.    SYNTHETIC  PROJECTIVE  GEOMETRY. 

By  George  Bruce  Halsted. 

No.  3.    DETERMINANTS. 

By  Laenas  Gifford  Weld. 

No.  4.    HYPERBOLIC   FUNCTIONS. 

By  James  McMahon. 

No.  5.    HARMONIC  FUNCTIONS. 

By  William  E.  Byerly. 

No.  6.    GRASSMANN'S  SPACE  ANALYSIS. 

By  Edward  W.  Hyde. 

No.  7.    PROBABILITY    AND  THEORY    OF   ERRORS. 

By  Robert  S.  Woodward. 

No.  8.    VECTOR  ANALYSIS  AND  QUATERNIONS. 

By  Alexander  Macfarlane. 

No.  9.    DIFFERENTIAL  EQUATIONS. 

By  William  Woolsey  Johnson. 

No.  10.  THE  SOLUTION  OF  EQUATIONS. 

By  Mansfield  Merriman. 

No.  II.    FUNCTIONS  OF  A  COMPLEX  VARIABLE. 

By  Thomas  S.  Fiske. 

PUBLISHED   BY 

JOHN  WILEY  &  SONS,   NEW  YORK. 
CHAPMAN  &  HALL,  Limited,  LONDON. 


MATHEMATICAL    MONOGRAPHS. 

EDITED    BY 

MANSFIELD   MERRIMAN  and  ROBERT   S.   WOODWARD. 


No.  8. 


VECTOR   ANALYSIS 


AND 


QUATERNIONS. 


BY 

ALEXANDER    MACFARLANE, 

Secretary  of  International  Association  for  Promoting  the  Study  of  Quaternions. 

BOSTON  COLLEGE  LIBKAK* 
CHESTNUT  HILL,  MASS. 

FOURTH    EDITION. 

FIRST    THOUSAND. 


mTH.  DEPT„ 

NEW  YORK: 

JOHN   WILEY    &    SONS. 

London:    CHAPMAN  &  HALL,    Limited. 

1906. 


Copyright,  1896, 

BY 

MANSFIELD   MERRIMAN  and  ROBERT  S.  WOODWARD 

UNDER   THE    TITLE 

HIGHER    MATHEMATICS. 

First  Edition,  September,  1896. 
Second  Edition,  January,  1898. 
Third  Edition,  August,  1900. 
Fourth  Edition,  January,  1906. 


1543 


ROBERT  DRUMMOND,    PRINTHR,    NEW   YORK. 


EDITORS'   PREFACE. 


The  volume  called  Higher  Mathematics,  the  first  edition 
of  which  was  published  in  1896,  contained  eleven  chapters  by 
eleven  authors,  each  chapter  being  independent  of  the  others, 
but  all  supposing  the  reader  to  have  at  least  a  mathematical 
training  equivalent  to  that  given  in  classical  and  engineering 
colleges.  The  publication  of  that  volume  is  now  discontinued 
and  the  chapters  are  issued  in  separate  form.  In  these  reissues 
it  will  generally  be  found  that  the  monographs  are  enlarged 
by  additional  articles  or  appendices  which  either  amplify  the 
former  presentation  or  record  recent  advances.  This  plan  of 
publication  has  been  arranged  in  order  to  meet  the  demand  of 
teachers  and  the  convenience  of  classes,  but  it  is  also  thought 
that  it  may  prove  advantageous  to  readers  in  special  lines  of 
mathematical  literature. 

It  is  the  intention  of  the  publishers  and  editors  to  add  other 
monographs  to  the  series  from  time  to  time,  if  the  call  for  the 
same  seems  to  warrant  it.  Among  the  topics  which  are  under 
consideration  are  those  of  elliptic  functions,  the  theory  of  num- 
bers, the  group  theory,  the  calculus  of  variations,  and  non- 
Euclidean  geometry;  possibly  also  monographs  on  branches  of 
astronomy,  mechanics,  and  mathematical  physics  may  be  included. 
It  is  the  hope  of  the  editors  that  this  form  of  publication  may 
tend  to  promote  mathematical  study  and  research  over  a  wider 
field  than  that  which  the  former  volume  has  occupied. 

December,  1905. 


AUTHOR'S   PREFACE. 


Since  this  Introduction  to  Vector  Analysis  and  Quaternions 
was  first  published  in  1896,  the  study  of  the  subject  has  become 
much  more  general;  and  whereas  some  reviewers  then  regarded 
the  analysis  as  a  luxury,  it  is  now  recognized  as  a  necessity  for 
the  exact  student  of  physics  or  engineering.  In  America,  Pro- 
fessor Hathaway  has  published  a  Primer  of  Quaternions  (New 
York,  1896),  and  Dr.  Wilson  has  amplified  and  extended  Pro- 
fessor Gibbs'  lectures  on  vector  analysis  into  a  text-book  for  the 
use  of  students  of  mathematics  and  physics  (New  York,  1901). 
In  Great  Britain,  Professor  Henrici  and  Mr.  Turner  have  pub- 
lished a  manual  for  students  entitled  Vectors  and  Rotors  (London, 
1903);  Dr.  Knott  has  prepared  a  new  edition  of  Kelland  and 
Tait's  Introduction  to  Quaternions  (London,  1904);  and  Pro- 
fessor Joly  has  realized  Hamilton's  idea  of  a  Manual  of  Quater- 
nions (London,  1905).  In  Germany  Dr.  Bucherer  has  pub- 
lished Elemente  der  Vektoranalysis  (Leipzig,  1903)  which  has 
now  reached  a  second  edition. 

Also  the  writings  of  the  great  masters  have  been  rendered 
more  accessible.  A  new  edition  of  Hamilton's  classic,  the  Ele- 
ments of  Quaternions,  has  been  prepared  by  Professor  Joly 
(London,  1899,  1901);  Tait's  Scientific  Papers  have  been  re- 
printed in  collected  form  (Cambridge,  1898,  1900);  and  a  com- 
plete edition  of  Grassmann's  mathematical  and  physical  works 
has  been  edited  by  Friedrich  Engel  with  the  assistance  of  several 
of  the  eminent  mathematicians  of  Germany  (Leipzig,  1894-). 
In  the  same  interval  many  papers,  pamphlets,  and  discussions 
have  appeared.  For  those  who  desire  information  on  the  litera- 
ture of  the  subject  a  Bibliography  has  been  published  by  the 
Association  for  the  promotion  of  the  study  of  Quaternions  and 
Allied  Mathematics  (Dublin,  1904). 

There  is  still  much  variety  in  the  matter  of  notation,  and  the 
relation  of  Vector  Analysis  to  Quaternions  is  still  the  subject 
of  discussion  (see  Journal  of  the  Deutsche  Mathematiker-Ver- 
einigung  for  1904  and  1905). 

Chatham,  Ontario,  Canada,  December,  1905. 


CONTENTS. 


Art.  i.  Introduction Page  7 

2.  Addition  of  Coplanar  Vectors 8 

3.  Products  of  Coplanar  Vectors 14 

4.  Coaxial  Quaternions 21 

5.  Addition  of  Vectors  in  Space 25 

6.  Product  of  Two  Vectors 26 

7.  Product  of  Three  Vectors 31 

8.  Composition  of  Located  Quantities 35 

9.  Spherical  Trigonometry 39 

10.  Composition  of  Rotations 45 


Index 


49 


VECTOR    ANALYSIS    AND    QUATERNIONS. 


Art.  1.    Introduction. 

By  "  Vector  Analysis  "  is  meant  a  space  analysis  in  which 
the  vector  is  the  fundamental  idea;  by  "  Quaternions"  is  meant 
a  space-analysis  in  which  the  quaternion  is  the  fundamental 
idea.  They  are  in  truth  complementary  parts  of  one  whole; 
and  in  this  chapter  they  will  be  treated  as  such,  and  developed 
so  as  to  harmonize  with  one  another  and  with  the  Cartesian 
Analysis.*  The  subject  to  be  treated  is  the  analysis  of  quanti- 
ties in  space,  whether  they  are  vector  in  nature,  or  quaternion 
in  nature,  or  of  a  still  different  nature,  or  are  of  such  a  kind  that 
they  can  be  adequately  represented  by  space  quantities. 

Every  proposition  about  quantities  in  space  ought  to  re- 
main true  when  restricted  to  a  plane ;  just  as  propositions 
about  quantities  in  a  plane  remain  true  when  restricted  to  a 
straight  line.  Hence  in  the  following  articles  the  ascent  to  the 
algebra  of  space  is  made  through  the  intermediate  algebra  of 
the  plane.  Arts.  2-4  treat  of  the  more  restricted  analysis, 
while  Arts.  5-10  treat  of  the  general  analysis. 

This  space  analysis  is  a  universal  Cartesian  analysis,  in  the 
same  manner  as  algebra  is  a  universal  arithmetic.  By  provid- 
ing an  explicit  notation  for  directed  quantities,  it  enables  their 
general  properties  to  be  investigated  independently  of  any 
particular  system  of  coordinates,  whether  rectangular,  cylin- 
drical, or  polar.     It  also  has  this  advantage  that  it  can  express 

*For  a  discussion  of  the  relation  of  Vector  Analysis  to  Quaternions,   see 
Nature,  1891-1893. 


8  VECTOR    ANALYSIS    AND    QUATERNIONS. 

the  directed  quantity  by  a  linear  function  of  the  coordinates, 
instead  of  in  a  roundabout  way  by  means  of  a  quadratic  func- 
tion. 

The  different  views  of  this  extension  of  analysis  which  have 
been  held  by  independent  writers  are  briefly  indicated  by  the 
titles  of  their  works  : 

Argand,  Essai  sur  une  maniere  de  representer  les  quantites 
imaginaires  dans  les  constructions  geornetriques,  1806. 

Warren,  Treatise  on  the  geometrical  representation  of  the  square 
roots  of  negative  quantities,  1828. 

Moebius,  Der  barycentrische  Calcul,  1827. 

Bellavitis,  Calcolo  delle  Equipollenze,  1835. 

Grassmann,  Die  lineale  Ausdehnungslehre,  1844. 

De  Morgan,  Trigonometry  and  Double  Algebra,  1849. 

O'Brien,  Symbolic  Forms  derived  from  the  conception  of  the 
translation  of  a  directed  magnitude.  Philosophical  Transactions, 
1851. 

Hamilton,  Lectures  on  Quaternions,  1853,  and  Elements  of 
Quaternions,  1866. 

Tait,  Elementary  Treatise  on  Quaternions,  1867. 

Hankel,  Vorlesungen  iiber  die  complexen  Zahlen  und  ihre 
Functionen,  1867. 

Schlegel,  System  der  Raumlehre,  1872. 

Houel,  Theorie  des  quantites  complexes,  1874. 

Gibbs,  Elements  of  Vector  Analysis,  188 1-4. 

Peano,  Calcolo  geometrico,  1888. 

Hyde,  The  Directional  Calculus,  1890. 

Heaviside,  Vector  Analysis,  in  "  Reprint  of  Electrical  Papers," 
1885-92. 

Macfarlane,  Principles  of  the  Algebra  of  Physics,  1891.  Papers 
on  Space  Analysis,  1891-3. 

An  excellent  synopsis  is  given  by  Hagen  in  the  second  volume 
of  his  "  Synopsis  der  hoheren  Mathematik." 

Art.  2.    Addition  of  Coplanar  Vectors. 

By  a  "  vector"  is  meant  a  quantity  which  has  magnitude 
and  direction.     It  is  graphically  represented  by  a  line  whose 


ADDITION    OF    COPLANAR    VECTORS.  V 

length  represents  the  magnitude  on  some  convenient  scale,  and 
whose  direction  coincides  with  or  represents  the  direction  of 
the  vector.  Though  a  vector  is  represented  by  a  line,  its 
physical  dimensions  maybe  different  from  that  of  a  line.  Ex- 
amples are  a  linear  velocity  which  is  of  one  dimension  in 
length,  a  directed  area  which  is  of  two  dimensions  in  length, 
an  axis  which  is  of  no  dimensions  in  length. 

A  vector  will  be  denoted  by  a  capital  italic  letter,  as  £*  its 
magnitude  by  a  small  italicletter,  as  b,  and  its  direction  by  a  small 
Greek  letter,  as  /?.  For  example,  B  =  bfi,  R  =  rp.  Sometimes 
it  is  necessary  to  introduce  a  dot  or  a  mark  /  to  separate 
the  specification  of  the  direction  from  the  expression  for  the 
magnitude  ;  f  but  in  such  simple  expressions  as  the  above,  the 
difference  is  sufficiently  indicated  by  the  difference  of  type.  A 
system  of  three  mutually  rectangular  axes  will  be  indicated, 
as  usual,  by  the  letters  i,j,  k. 

The  analysis  of  a  vector  here  supposed  is  that  into  magni- 
tude and  direction.  According  to  Hamilton  and  Tait  and 
other  writers  on  Quaternions,  the  vector  is  analyzed  into  tensor 
and  unit-vector,  which  means  that  the  tensor  is  a  mere  ratio 
destitute  of  dimensions,  while  the  unit-vector  is  the  physical 
magnitude.  But  it  will  be  found  that  the  analysis  into  magni- 
tude and  direction  is  much  more  in  accord  with  physical  ideas, 
and  explains  readily  many  things  which  are  difficult  to  explain 
by  the  other  analysis. 

A  vector  quantity  may  be  such  that  its  components  have  a 
common  point  of  application  and  are  applied  simultaneously; 
or  it  may  be  such  that  its  components  are  applied  in  succes- 
sion, each  component  starting  from  the  end  of  its  predecessor. 
An  example  of  the  former  is  found  in  two  forces  applied  simul- 
taneously at  the  same  point,  and  an  example  of  the  latter  in 

*This  notation  is  found  convenient  by  electrical  writers  in  order  to  harmo- 
nize with  the  Hospitalier  system  of  symbols  and  abbreviations. 

f  The  dot  was  used  for  this  purpose  in  the  author's  Note  on  Plane  Algebra, 
1883;  Kennelly  has  since  used  Z  for  the  same  purpose  in  his  electrical  papers 


10  VECTOR    ANALYSIS    AND    QUATERNIONS. 

two  rectilinear  displacements  made  in  succession  to  one  an- 
other. 

Composition  of  Components  having  a  common  Point  of 
Application. — Let  OA  and  OB  represent  two  vectors  of  the 
same  kind  simultaneously  applied  at  the  point  O.  Draw  BC 
B  c  parallel  to  OA,  and  AC  parallel  to  OB,  and 

join  OC.     The  diagonal  OC  represents  in  mag- 
nitude and  direction  and  point  of  application 
o  ~^a      the  resultant  of  OA  and  OB.     This  principle 

was  discovered  with  reference  to  force,  but  it  applies  to  any 
vector  quantity  coming  under  the  above  conditions. 

Take  the  direction  of  OA  for  the  initial  direction  ;  the  di- 
rection of  any  other  vector  will  be  sufficiently  denoted  by  the 
angle  round  which. the  initial  direction  has  to  be  turned  in 
order  to  coincide  with  it.  Thus  OA  may  be  denoted  by 
fjo,  OB  by/2/^,  OC  by//0.  From  the  geometry  of  the  fig- 
ure it  follows  that 

f^/S+f;  +  2A/.COS*. 


and  tan  0  = 


/.-h/.cos*; 


/,  sin  0S 


hence  OC  =  Vf*  +  /a2  +  2fJ,  cos  V,  /tan^  ^  ^  &. 

Example. — Let  the  forces  applied  at  a  point  be  2/00  and 

3/600.      Then    the  resultant  is  ^4  +  9+  I2X|  /tarr1  - — ^ 
=  4.36/36!^. 

If  the  first  component  is  given  as/,/0,,  then  we  have  the 
more  symmetrical    formula 

OC  =  •/,'+/.■+ */,/.  co,  (#..-0.)  l^JZl'XiZX- 

When  the  components  are  equal,  the  direction  of  the  re- 
sultant bisects  the  angle  formed  by  the  vectors ;  and  the  mag- 
nitude of  the  resultant  is  twice  the  projection  of  either  compo- 
nent on  the  bisecting  line.     The  above  formula  reduces  to 

OC  =  2/  cos  4  ld-\ 
2/  2 


ADDITION    OF   COPLANAR   VECTORS.  11 

Example. — The  resultant  of  two  equal  alternating  electro- 
motive forces  which  differ  1200  in  phase  is  equal  in  magnitude 
to  either  and  has  a  phase  of  6o°. 

Given  a  vector  and  one  component,  to  find  the  other  com- 
ponent.— Let  OC  represent  the  resultant,  and  OA  the  compo- 
nent. Join  AC  and  draw  OB  equal  and  B  c 
parallel  to  AC.  The  line  OB  represents 
the  component  required,  for  it  is  the  only  ,/' 
line  which  combined  with  OA  gives  OC  A'  o  ~*A- 
as  resultant.  The  line  OB  is  identical  with  the  diagonal  of  the 
parallelogram  formed  by  OC  and  OA  reversed  ;  hence  the  rule 
is,  "  Reverse  the  direction  of  the  component,  then  compound 
it  with  the  given  resultant  to  find  the  required  component." 
Let  f/0  be  the  vector  and  fjo  one  component ;  then  the 
■other  component  is 

m = *y-+/.-  -  *//.  cos  »Mn_/+n/C0S  9 

Given  the  resultant  and  the  directions  of  the  two  compo- 
nents, to  find  the  magnitude  of  the  components. — The  resultant 
is  represented  by  OC,  and  the  directions  by  OX  and  OY. 
Yr  From   C  draw  CA  parallel  to  OY,  and  CB 

parallel  to  OX ;  the  lines  OA  and  OB  cut 
off  represent  the  required  components.  It 
is  evident  that  OA  and  OB  when  com- 
pounded  produce  the  given  resultant  OC, 
and  there  is  only  one  set  of  two  components  which  produces 
a  given  resultant ;  hence  they  are  the  only  pair  of  components 
having  the  given  directions. 

Let//0  be  the  vector  and  /dl  and  /di  the  given  directions. 

Then 

/  +/.  cos  (0,  -  0t)  =/cos  (d  -  0,), 

fx  cos  (0,  -  6,)  +/.  =  /cos  (0,  -  8), 
from  which  it  follows  that 

{cos  (6  -  0.)  -  cos  (0,  -  ff)  cos  (0,  -  0,)  } 
7l      J  i  -cosa(02-  0,) 


12  VECTOR    ANALYSIS    AND    QUATERNIONS. 

For  example,  let  100/600,  /300,  and  /900  be  given  ;  then 
.  cos  300 

A  =  100 


1  -(-  cos  6o°* 

Composition  of  any  Number  of  Vectors  applied  at  a  com- 
mon Point. — The  resultant  may  be  found  by  the  following 
graphic  construction  :  Take  the  vectors  in  any  order,  as  A,  B,  C. 
From  the  end  of  A  draw  B'  equal  and  par- 
allel to  B,  and  from  the  end  of  B'  draw  C 
)B  equal  and  parallel  to  C\  the  vector  from 
the  beginning  of  A  to  the  end  of  C  is  the 
resultant  of  the  given  vectors.  This  follows 
~a ^  by  continued  application  of  the  parallelo- 
gram construction.  The  resultant  obtained  is  the  same,  what- 
ever the  order;  and  as  the  order  is  arbitrary,  the  area  enclosed 
has  no  physical  meaning. 

The  result  may  be  obtained  analytically  as  follows : 
Given  A/0,  +  /2/0a  +  fj±%  +  .  .  .  +  /,  /0„ 

Now  fj\  =/lCos  »1/o+/1sin  Qx[\ 

Similarly  fJA  =  /.  cos  BJo +/.  sin  ,2/| 

lit 
and  fn/On  =  f«  cos  0n/o  ~\-fn  sin  0„  /  — . 

Hence  2{//d\  =  |^/cos  0}  /o  +  {^/sin  #}    /j 

=  |/(5/cos6/)«+(2/sin6f)'  .  tan-1^^^- 

In  the  case  of  a  sum  of  simultaneous  vectors  applied  at  a  com- 
mon point,  the  ordinary  rule  about  the  transposition  of  a  term  in 
an  equation  holds  good.  For  example,  if  A  -\-B  +  C  —  o,  then 
A  +  B  -  ~  C,  and  A  +  C  =  -  £,  and  B  +  C  =  -  A,  etc. 
This  is  permissible  because  there  is  no  real  order  of  succession 
among  the  given  components.* 

*  This  does  not  hold  true  of  a  sum  of  vectors  having  a  real  order  of  succes- 
sion.    It  is  a  mistake  to  attempt  to  found  space-analysis  upon  arbitrary  formal 


ADDITION    OF    COPLANAR    VECTORS.  IS 

Composition  of  Successive  Vectors. — The  composition  of 
successive  vectors  partakes  more  of  the  nature  of  multiplica- 
tion than  of  addition.  Let  A  be  a  vector  start-  A 
ing  from  the  point  O,  and  B  a  vector  starting 
from  the  end  of  A.  praw  the  third  side  OP, 
and  from  O  draw  a  vector  equal  to  B,  and  from 
its  extremity  a  vector  equal  to  A.  The  line  OP  is  not  the 
complete  equivalent  of  A  -f-  B ;  if  it  were  so,  it  would  also  be 
the  complete  equivalent  of  B  -{-  A.  But  A  -\-  B  and  B  -\-A 
determine  different  paths;  and  as  they  go  oppositely  around, 
the  areas  they  determine  with  OP  have  different  signs.  The 
diagonal  OP  represents  A  -\-  B  only  so  far  as  it  is  consid- 
ered independent  of  path.  For  any  number  of  successive 
vectors,  the  sum  so  far  as  it  is  independent  of 
path  is  the  vector  from  the  initial  point  of  the 
first  to  the  final  point  of  the  last.  This  is  also 
true  when  the  successive  vectors  become  so  small 
as  to  form  a  continuous  curve.  The  area  between 
the  curve  OPQ  and  the  vector  OQ  depends  on  the  path,  and 
has  a  physical  meaning. 

Prob.  i.  The  resultant  vector  is  1 23/45 °,  and  one  component 
is  100/00;  find  the  other  component. 

Prob.  2.  The  velocity  of  a  body  in  a  given  plane  is  200  /750,  and 
one  component  is  100/250;  find  the  other  component. 

Prob.  3.  Three  alternating  magnetomotive  forces  are  of  equal 
virtual  value,  but  each  pair  differs  in  phase  by  1200;  find  the  re- 
sultant.    (Ans.  Zero.) 

Prob.  4.  Find  the  components  of  the  vector  100/700  in  the  direc- 
tions 200  and  ioo°. 

Prob.  5.  Calculate  the  resultant  vector  of  1/100,  2/200,  3/300, 
4/40°- 

Prob.  6.  Compound  the  following  magnetic  fluxes:  h  sin  ///  -)- 
h  sin  {nt  —  i2o°)/i2o°  +  h  sin  (nt  —  240°)/24o°.      (Ans.  §/z int.) 

laws;  the  fundamental  rules  must  be  made  to  express  universal  properties  of  the 
thing  denoted.  In  this  chapter  no  attempt  is  made  to  apply  formal  laws  to 
directed  quantities.     What  is  attempted  is  an  analysis  of  these  quantities. 


14  VECTOR    ANALYSIS    AND    QUATERNIONS. 

Prob.  7.  Compound  two  alternating  magnetic  fluxes  at  a  point, 

a  cos  nt  /o  and  a  sin  nt  /— .      (Ans.  a  /fit.) 
-  /     2 

Prob  8.  Find  the  resultant  of  two  simple  alternating  electromo- 
tive forces  100/200  and  50/750. 

Prob.  9.  Prove  that  a  uniform  circular  motion  is  obtained  by 
compounding  two  equal  simple  harmonic  motions  which  have  the 
space-phase  of  their  angular  positions  equal  to  the  supplement  of  the 
time-phase  of  their  motions. 

Art.  3.    Products  of  Coplanar  Vectors. 

When  all  the  vectors  considered  are  confined  to  a  common 
plane,  each  may  be  expressed  as  the  sum  of  two  rectangular 
components.  Let  i  and/  denote  two  directions  in  the  plane  at 
right  angles  to  one  another  ;  then  A  =  axi  -j-  a  J,  B  =  bxi  -f-  bj, 
R  =  xi-\-yj.  Here  i  and  j  are  not  unit-vectors,  but  rather 
signs  of  direction. 

Product  of  two  Vectors.  — Let  A  =  aj-\-aJ  and  B  =  bti-\-bJ 
be  any  two  vectors,  not  necessarily  of  the  same  kind  physically. 
We  assume  that  their  product  is  obtained  by  applying  the 
distributive  law,  but  we  do  not  assume  that  the  order  of  the 
factors  is  indifferent.  Hence 
AB  =  {axi  +  aJ){bj-\-  bj)  =  afiji  +  aJ?Jj-\-  axbjj-\-a%bJi. 

If  we  assume,  as  suggested  by  ordinary  algebra,  that  the 
square  of  a  sign  of  direction  is  -j-,  and  further  that  the  product 
of  two  directions  at  right  angles  to  one  another  is  the  direction 
normal  to  both,  then  the  above  reduces  to 

AB  —  axbx  +  tf  A  +  (« A  —  aj)x)k. 

Thus  the  complete  product  breaks  up  into  two  partial 
products,  namely,  axbx  -J-  aj)%  which  is  independent  of  direc- 
tion, and  (axb2 — aj?x)k  which  has  the  axis  of  the  plane  for 
direction.* 

*  A  common  explanation  which  is  given  of  ij  =  k  is  that  i  is  an  operator,/ an 
operand,  and  k  the  result.  The  kind  of  operator  which  i  is  supposed  to  denote 
is  a  quadrant  of  turning  round  the  axis  i  ;  it  is  supposed  not  to  be  an  axis,  but 
a  quadrant  of  rotation  round  an  axis.  This  explains  the  result  ij  =  k,  but 
unfortunately  it  does  not  explain  ii  —  -f-  ;  for  it  would  give  ii  =  i. 


PRODUCTS  OF  COPLANAR  VECTORS. 


15 


Scalar  Product  of  two  Vectors. — By  a  scalar  quantity  is 
meant  a  quantity  which  has  magnitude  and  may  be  positive  or 
negative  but  is  destitute  of  direction.  The  former  partial 
product  is  so  called  because  it  is  of  such  a  nature.  It  is 
denoted  by  SAB  where  the  symbol  S,  being  in  Roman  type, 
denotes,  not  a  vector,  but  a  function  of  the 
vectors  A  and  B.  The  geometrical  mean- 
ing of  SAB  is  the  product  of  A  and  the 
orthogonal  projection  of  B  upon  A.  Let 
OP  and  OQ  represent  the  vectors  A  and  B; 
draw   QM  and  NL  perpendicular  to    OP. 


Then 


a 


(OP)(OM)  =  (OP)(OL)  +  (OP)(LM), 


=  a  <  b. r  K~\* 

(    'a     '      2  a ) 


=  alb1-\-  a&. 

Corollary  I. — SB  A  =  SAB.  For  instance,  let  A  denote  a 
force  and  B  the  velocity  of  its  point  of  application  ;  then  SAB 
denotes  the  rate  of  working  of  the  force.  The  result  is  the 
same  whether  the  force  is  projected  on  the  velocity  or  the 
velocity  on  the  force. 

Example  I. — A  force  of  2  pounds  East  -(-  3  pounds  North  is 
moved  with  a  velocity  of  4  feet  East  per  second  -|-  5  feet  North 
per  second ;  find  the  rate  at  which  work  is  done. 

2X4+3X5=23  foot-pounds  per  second. 

Corollary  2. — A"1  =*  a?  -f-  a*  —  a2.  The  square  of  any  vector 
is  independent  of  direction ;  it  is  an  essentially  positive  or 
signless  quantity  ;  for  whatever  the  direction  of  A,  the  direction 
of  the  other  ^4  must  be  the  same;  hence  the  scalar  product 
cannot  be  negative. 

Example  2. — A  stone  of  10  pounds  mass  is  moving  with  a 
velocity  64  feet  down  per  second  -j-  100  feet  horizontal  per 
second.    Its  kinetic  energy  then  is 

—  (64'+  ioo")  foot-poundals, 


16  VECTOR    ANALYSIS    AND    QUATERNIONS. 

a  quantity  which  has  no  direction.  The  kinetic  energy  due  to 

64* 

the  downward  velocity  is  10  X  —  and  that  due  to  the  hori- 
zontal velocity  is  —  X  ioo2  ;  the  whole  kinetic  energy  is  ob- 
tained, not  by  vector,  but  by  simple  addition,  when  the  com- 
ponents are  rectangular. 

Vector  Product  of  two  Vectors. — The  other  partial  product 

from  its  nature  is  called  the  vector  product,  and  is  denoted  by 

VAB.      Its    geometrical    meaning    is    the 

product  of  A  and  the  projection  of  B  which 

is  perpendicular  to  A,  that  is,  the  area  of 

the  parallelogram  formed  upon  A  and  B. 

Let  OP  and  OO  represent  the  vectors  A 

and  B,  and  draw  the  lines  indicated  by  the 

figure.       It  is  then  evident  that  the  area 

of  the  triangle  OPQ  =  aj>%  —  ^atqt  —  \bxb^  —  \{ax  —  b,){b.2  —  a,), 

=  \{aj),  -  aj)x). 

Thus  (#i#9  —  ajb^)k  denotes  the  magnitude  of  the  parallelo- 
gram formed  by  A  and  B  and  also  the  axis  of  the  plane  in 
which  it  lies. 

It  follows  that  MBA  =  —  VAB.  It  is  to  be  observed 
that  the  coordinates  of  A  and  B  are  mere  component  vectors, 
whereas  A  and  B  themselves  are  taken  in  a  real  order. 

Example. — Let  A  =  (loi -\-  nj)  inches  and  B  =  (52  +  127*) 
inches,  then  VAB  =  (120—  $$)k  square  inches;  that  is,  65 
square  inches  in  the  plane  which  has  the  direction  k  for  axis. 

If  A  is  expressed  as  aa  and  B  as  bft,  then  SAB  =  ab  cos  aft, 
where  a/3  denotes  the  angle  between  the  directions  a  and  ft. 

Example. — The  effective  electromotive  force  of  100  volts 
per  inch  /900  along  a  conductor  8  inch  /450  is  SAB  =  8  X  100 
cos/450  /900  volts,  that  is,  800  COS450  volts.  Here/450  indicates 
the  direction  a  and  /900  the  direction  (3,  and  /450  /900  means 
the  angle  between  the  direction  of  450  and  the  direction  of  900. 

Also  VAB  =  ab  sin  aft  .  ~aft,  where  aft  denotes  the  direction 
which  is  normal  to  both  a  and  ft,  that  is,  their  pole. 


PRODUCTS  OF  COPLANAR  VECTORS.  17 

Example. — At  a  distance  of  10  feet  /300  there  is  a  force  of 
IOO  pounds  /6o°.     The  moment  is  VAB 

=  10  X  100  sin  /300  /6o°  pound-feet  900/  /900 
=  1000  sin  300  pound-feet  900/  /900. 
Here  900/  specifies  the  plane  of  the  angle  and  /900  the  angle. 
The  two  together  written  as  above  specify  the  normal  k. 

Reciprocal  of  a  Vector. — By  the  reciprocal  of  a  vector  is 
meant  the  vector  which  combined  with  the  original  vector  pro- 
duces the  product  -j-  1.  The  reciprocal  of  A  is  denoted 
by  A~\  Since  AB  =  ab  (cos  a/3  -f-  sin  a/3 .  a/3),  b  must  equal 
^r1  and  /3  must  be  identical  with  a  in  order  that  the  product 
may  be  1.     It  follows  that 

_  1  aa       aj  +  aj 

J~L        —        a  —       —  —         -      ;  — • 

a         a         a,   -j-  a2 

The  reciprocal  and  opposite  vector  is  —  A~l.     In  the  figure 
let  OP  =  2/3  be  the  given  vector ;  then  OQ  =  \f$  is  its  recipro- 
cal, and  OR  =  \ (  —  /3)  is  its  reciprocal  and 
opposite.*  R         Q  p 

Example.— If  A  =   10  feet  East  -f  5    feet  North,  A~x  = 

feet     East  +   —    feet    North    and  —  A~l  =  feet 

125  '     125  125 

East   —    —  feet  North. 
125 

Product  of  the  reciprocal  of  a  vector  and  another  vector. — 
A-'B  =  \AB, 

(X 

=  ~2  \<*A  +  "A  +  (« A  —  aA)<*fi}> 

b  f  .         — 

=  -  (cos  a/3  -j-  sin  a/3 .  a/3). 


*  Writers  who  identify  a  vector  with  a  quadrantal  versor  are  logically  led  to 
define  the  reciprocal  of  a  vector  as  being  opposite  in  direction  as  well  as  recip- 
rocal in  magnitude. 


18  VECTOR    ANALYSIS    AND    QUATERNIONS. 

h  h 

Hence  SA~*B  =  -cos  a  8  and  VA~lB  =  -sin  a8.a8. 
a  a  ' 

Product  of  three  Coplanar  Vectors. — Let  A  =  aj  +  a^j, 

B  =  bj  -f-  bj,     C  =  cj  -f-  c9j  denote  any  three  vectors  in  a 

common  plane.     Then 

(AB)C  =  {{aA  +  a%b%)  +  ifiA  -  afak}{cxi  +  cj) 

=  (a A  +  «A)fo*  +  CJ)  +  («A  -  «A)(—  V  +  ^/  )• 
The  former  partial  product  means  the  vector  £7  multiplied 
by  the  scalar  product  of  A  and  B  ;  while  the 
latter    partial    product    means  the  comple- 
mentary vector  of  C  multiplied  by  the  mag- 
nitude of  the  vector  product  of  A   and  B. 
"o~~         — J    If  these  partial  products  (represented  by  OP 
and  00)  unite  to  form  a  total  product,  the  total  product  will  be 
represented  by  OR,  the  resultant  of  OP  and  OQ. 

The  former  product  is  also  expressed  by  SAB  .  C,  where  the 
point  separates  the  vectors  to  which  the  S  refers  ;  and  more 
analytically  by  abc  cos  aft  .  y. 

The  latter  product  is  also  expressed  by  (VAB)C,  which  is 
equivalent    to    V(VAB)C,   because   VAB  is  at    right    angles 

to  C.  It  is  also  expressed  by  abc  sin  at/3 .  afiy,  where  afSy  de- 
notes the  direction  which  is  perpendicular  to  the  perpendicular 
to  a  and  (3  and  y. 

If  the  product  is  formed  after  the  other  mode  of  association 
we  have 
A{BC)  =  (aJ-\-  aj){b,cx  +  V.)  +  (*.*  +  **/)(V.  -  K^k 
=  {b,cx  +  b2c,)(aj  +  a  J)  +  {b,c2  -  b.c^aj  -  a  J) 
=  SBC.A  +VA(VBC). 

The  vector  aj  —  a  J  is  the  opposite  of  the  complementary 
-vector  of  aj,  -f-  a* J-     Hence  the  lattei  partial  product  differs 
with  the  mode  of  association. 

Example.— Let   A  (  =  i/o  +  2/900,     B  =  3/00  +  4/90° , 
C  =  5/00  -\-  6/900.     The  fourth  proportional  to  A,  B,  C  is 


PRODUCTS  OF  COPLANAR  VECTORS.  19> 


(A-*B)C=  I  XT3+'X4!5/g!  +  6/9Q°  } 

I     -j-  2 


1X4-2X3 


{  -6/o°  +  $/9o°\ 


I' +  2' 

=   134/0^+11.2/90°. 

Square  of  a  Binomial  of  Vectors. — If  A  -\-  B  denotes  a 
sum  of  non-successive  vectors,  it  is  entirely  equivalent  to  the 
resultant  vector  C.  But  the  square  of  any  vector  is  a  positive 
scalar,  hence  the  square  of  A  -\-  B  must  be  a  positive  scalar. 
Since  A  and  B  are  in  reality  components  of  one  vector,  the 
square  must  be  formed  after  the  rules  for  the  products  of  rect- 
angular components  (p.  432).     Hence 

(A  +B)2  =  {A  +  B){A  +  B), 

=  A'  +  AB  +  BA  +  B\ 

=  A2  +  B1  +  SAB  +  SB  A  +  VAB  +  NBA, 

=  A*  +  B*  +  2SAB. 

This  may  also  be  written  in  the  form 

a2  -f-  b2  -f  2ab  cos  a/3. 

But  when  A  -\-  B  denotes  a  sum  of  successive  vectors,  there 
is  no  third  vector  C  which  is  the  complete  equivalent  ;  and  con- 
sequently we  need  not  expect  the  square  to  be  a  scalar  quan- 
tity. We  observe  that  there  is  a  real  order,  not  of  the  factors, 
but  of  the  terms  in  the  binomial;  this  causes  both  product 
terms  to  be  AB,  giving 

{A  +  B)2=  A%  +  2AB  +  B2 

=  A2+B2  +  2SAB  +  2VAB. 
The  scalar  part  gives  the  square  of  the  length  of  the  third 
side,  while  the  vector  part  gives  four  times  the  area  included 
between  the  path  and  the  third  side. 

Square  of  a  Trinomial  of  Coplanar  Vectors. — Let  .4  -J-  B  -j- 
C  denote  a  sum  of  successive  vectors.  The  product  terms  must 
be  formed  so  as  to  preserve  the  order  of  the  vectors  in  the  tri- 
nomial ;  that  is,  A  is  prior  to  B  and   C,  and  B  is  prior  to  C. 


20  VECTOR    ANALYSIS    AND    QUATERNIONS. 

Hence 

(A  +  B  +  Cf  =  A'  4-  B*  +  C  +  2AB -\-2AC-\-  2BC, 

=  a2  +  ^2  +  c2  +  2(s^^  +  s^r  +  s^o,  (1) 

4-  2(V^^  +  VA  C  +  V£Q.  (2) 

Hence  S(^+^+Q2=  (1) 

=  a2  -f-  £9  +  ^2  +  2<a:^  cos  a/^  +  2ac  cos  ^T  +  2^  cos  @Y 

and  V(^+^+6:)2  =  (2) 

=  {2^  sin  a/3  -f-  2«£  sin  a^  -}-  2^  sin  /fy/f.  afi 

The  scalar  part  gives  the  square  of  the  vector  from  the  be- 
c     ginning  of  A  to  the  end  of  C  and  is  all  that  exists 
when  the  vectors  are  non-successive.     The  vector 
g  part  is  four  times  the  area  included  between  the 
successive   sides   and   the   resultant   side  of   the 
a  polygon. 

Note  that  it  is  here  assumed  that  V(A  +  B)C '=  VAC-\- 
VBC,  which  is  the  theorem  of  moments.  Also  that  the  prod- 
uct terms  are  not  formed  in  cyclical  order,  but  in  accordance 
with  the  order  of  the  vectors  in  the  trinomial. 

Example.— Let  A  =  3/^  B  =  5/300,  C  =  7/45°  ;  find  the 
area  of  the  polygon. 

±V{AB+  AC  +  BC), 
=  i{i5sin/o/30°  +  2i  sin/o/45°  +  35  sin /300 /450}, 

=  375  +  742  +  4-53  =  157- 

Prob.  10.  At  a  distance  of  25  centimeters  /200  there  is  a  force 
of  1000  dynes  /8o°;  find  the  moment. 

Prob.  11.  A  conductor  in  an  armature  has  a  velocity  of  240 
inches  per  second  /3000  and  the  magnetic  flux  is  50,000  lines  per 
square  inch  /o;  find  the  vector  product. 

(Ans.   1.04  X  ioT  lines  per  inch  per  second.) 

Prob.  12.  Find  the  sine  and  cosine  of  the  angle  between  the 
directions  0.8141  E.  +  0.5807  N.,  and  0.5060  E.  +  0.8625  N. 

Prob.  13.  When  a  force  of  200  pounds  /2700  is  displaced  by 
10  feet  /300,  what  is  the  work  done  (scalar  product)  ?  What  is  the 
meaning  of  the  negative  sign  in  the  scalar  product  ? 


COAXIAL    QUATERNIONS.  21 

Prob.  14.  A  mass  of  ioo  pounds  is  moving  with  a  velocity  of  30 
feet  E.  per  second  -f-  50  feet  SE.  per  second;  find  its  kinetic  energy. 

Prob.  15.  A  force  of  10  pounds  /450  is  acting  at  the  end  of  8 
feet  /2000;  find  the  torque,  or  vector  product. 

Prob.  16.  The  radius  of  curvature  of  a  curve  is  2/00  +  5/900; 
find  the  curvature.     \  (Ans.  .03/0°  -f-  .17/900.) 

Prob.  17.  Find  the  fourth  proportional  to  10/00  +  2/900 
8/o°  -  3/9o_°,  and  6/o_°  +  5/V°. 

Prob.  18.  Find  the  area  of  the  polygon  whose  successive  sides 
are  10/300,  9/1000,  8/1800,  7/2250. 

Art.  4.    Coaxial  Quaternions. 

By  a  "  quaternion  "  is  meant  the  operator  which  changes 
one  vector  into  another.  It  is  composed  of  a  magnitude  and 
a  turning  factor.  The  magnitude  may  or  may  not  be  a  mere 
ratio,  that  is,  a  quantity  destitute  of  physical  dimensions ;  for 
the  two  vectors  may  or  may  not  be  of  the  same  physical  kind. 
The  turning  is  in  a  plane,  that  is  to  say,  it  is  not  conical.  For 
the  present  all  the  vectors  considered  lie  in  a  common  plane ; 
hence  all  the  quaternions  considered  have  a  common  axis.* 

Let  A  and  R  be  two  coinitial  vectors  ;  the  direction  normal 
to  the  plane  may  be  denoted   by  /?.     The   operator   which 
changes  A  into  R  consists  of  a  scalar  multiplier 
and  a  turning  round  the  axis  /?.    Let  the  former  be 
denoted  by  r  and  the  latter  by  fie,  where  6  denotes 
the  angle  in  radians.      Thus  R  =  rfPA  and  recip- 
rocally A  =  -6-dR.     Also  Lr  =  r66  and  ±A  =  -B~6. 
3  f>  A  r  R  rr 

The  turning  factor  ft9  may  be  expressed  as  the  sum  of  two 
component  operators,  one  of  which  has  a  zero  angle  and  the 
other  an  angle  of  a  quadrant.     Thus 

0°  =  cos  6  .  /3°  +  sin  6  .  p'\ 

*  The  idea  of  the  "quaternion  "  is  due  to  Hamilton.  Its  importance  may 
be  judged  from  the  fact  that  it  has  made  solid  trigonometrical  analysis  possible. 
It  is  the  most  important  key  to  the  extension  of  analysis  to  space.  Etymologi- 
cally  "quaternion"  means  denned  by  four  elements;  which  is  true  in  space  •  in 
plane  analysis  it  is  denned  by  two. 


22  VECTOR    ANALYSIS    AND    QUATERNIONS. 

When  the  angle  is  naught,  the  turning-factor  may  be 
omitted ;  but  the  above  form  shows  that  the  equation  is 
homogeneous,  and  expresses  nothing  but  the  equivalence  of  a 
given  quaternion  to  two  component  quaternions.* 

Hence  rfi9  =  r  cos  0  -\-r  s\r\  0  .  fi"'2 

and  r/3'A  =  pA  -\-qp«l*A 

=  pa  .  a  -\-  qa  .  fin/2a. 
The  relations  between  r  and  6,  and  p  and  q,  are  given  by 

r  =  Vf~+7,    e  =  tan  -xt 

q 

Example. — Let  E  denote  a  sine  alternating  electromotive 
force  in  magnitude  and  phase,  and  /  the  alternating  current  in 
magitude  and  phase,  then 

E  =  (r  +  27tnl .  fi"/2)/, 
where  r  is  the  resistance,  /  the  self-induction,  n  the  alternations 
per  unit  of  time,  and  /?  denotes  the  axis  of  the  plane  of  repre- 
sentation.    It  follows  that  E  =  rl  -\-  2-nnl .  fin/2f;  also  that 

I-lE  =  r-\-27t1ll.  /S^2, 
that  is,  the  operator  which  changes  the  current  into  the  elec- 
tromotive force  is  a  quaternion.     The  resistance  is  the  scalar 
part  of  the  quaternion,  and  the  inductance  is  the  vector  part. 
Components  of   the    Reciprocal  of   a  Quaternion. — Given 
•R=(p  +  g.p"*)A, 

then  A=P  +  9-P">R 

p-q.  F*  R 


{P  +  q.p"*)(p-q.p"<) 
R 


p— q  .  /3n/z 


*  In  the  method  of  complex  numbers  ^/a  is  expressed  by  i,  which  stands 
for  \/  —  i.  The  advantages  of  using  the  above  notation  are  that  it  is  capable 
of  being  applied  to  space,  and  that  it  also  serves  to  specify  the  general  turning 
factor  yffe  as  well  as  the  quadrantal  turning  factor  fi*fr. 


COAXIAL   QUATERNIONS.  23' 

Example. — Take  the  same  application  as  above.  It  is  im- 
portant to  obtain  /  in  terms  of  E.  By  the  above  we  deduce 
that  from  E  =  (r  -f-  2rrnl .  /3"/2)I 

/=  | r- 2nnl         6^\e 

Addition  of  Coaxial  Quaternions. — If  the  ratio  of  each  of 
several  vectors  to  a  constant  vector  A  is  given,  the  ratio  of 
their  resultant  to  the  same  constant  vector  is  obtained  by  tak- 
ing the  sum  of  the  ratios.     Thus,  if 

Rn  =    (A  +  9n  ■  ^)A, 

then  2  R  =  { 2p  +  (2g) .  ^}A, 

and  reciprocally 

A_2p-  {2g) .  F» 

(2py  +  {2gy   *•"- 

Example. — In  the  case  of  a  compound  circuit  composed 
of  a  number  of  simple  circuits  in  parallel 

_  r,-2nnl,.pl*  r9  -  2nnl9 .  (3^ 

Jl~  r?  +  {2itnyi?     '  2~~    rJ  +  ^Ttnyi?11"   ClC'> 

therefore,  21  —  2  \     „   ,    , — '-^rw  X  E 

{   r  +(27r/z)2/2   ) 

[        \r  -\-{27t7iyr)  r  -\-{27tnyi2  r      j 

and  reciprocally 

^(    ,    ,    / rra)  +  2  **^(— _, _)  .  /J»/i 

^  V  +  (27r«)/V    '  \r2-\-(27tnyPJ 

E  =  i —  v r^i — y—2/* 

\2r*  +  {2nnyr)  +  ^^  ^V  +  ^V'J 

Product  of  Coaxial  Quaternions. — If  the  quaternions  which 

change  ^4   to  7?,  and  R  to  i?',  are  given,  the  quaternion  which 

changes  A  to  R'  is  obtained  by  taking  the  product  of  the  given 

quaternions. 

*This  theorem  was  discovered  by  Lord  Rayleigh;   Philosophical  Magazine, 
May,  1886.     See  also  Bedell  &  Crehore's  Alternating  Currents,  p.  238. 


24  VECTOR    ANALYSIS   AND    QUATERNIONS. 

Given  R  =  r/3°A  =  {p-\-q.  fi"/z)A 

and  R!  =  r' §»' R  =  (/  +  q' .  P*'*)R, 

then  R'  =  rS  jP+*'  A  ={{pp'  -  qq')  +  (/?'  +p'q) .  /3"/*}A. 

Note  that  the  product  is  formed  by  taking  the  product  of 
the  magnitudes,  and  likewise  the  product  of  the  turning  fac- 
tors. The  angles  are  summed  because  they  are  indices  of  the 
common  base  /?.* 

Quotient  of  two  Coaxial  Quaternions. — If  the  given  qua- 
ternions are  those  which  change  A  to  R,  and  A  to  R',  then  that 
which  changes  R  to  Rf  is  obtained  by  taking  the  quotient  of 
the  latter  by  the  former. 

Given    R  =  r/36A  =  {p  -f-  q  .  ft"/*)A 
and  R!  =  r'j3<>'A  =  \p'  -f  q' .  ^)At 

then  R' =  r-f5»'-»R, 

r 

._  (PP'  +  qf)  +  (pg'-p'q).p",p 

p'  +  e' 

Prob.  19.  The  impressed  alternating  electromotive  force  is  200 
volts,  the  resistance  of  the  circuit  is  10  ohms,  the  self-induction  is 
•3-ro  henry,  and  there  are  60  alternations  per  second  ;  required  the 
current.  (Ans.   18.7  amperes  /—  200  42'.) 

Prob.  20.  If  in  the  above  circuit  the  current  is  10  amperes,  find 
the  impressed  voltage. 

Prob.  21.  If  the  electromotive  force  is  no  volts  /0  and  the  cur- 
rent is  10  amperes  /B  —  \n,  find  the  resistance  and  the  self-induc- 
tion, there  being  1 20  alternations  per  second. 

Prob.  22.  A  number  of  coils  having  resistances  rv  rv  etc.,  and 
self-inductions  lt ,  /2 ,  etc.,  are  placed  in  series  ;  find  the  impressed 
electromotive  force  in  terms  of  the  current,  and  reciprocally. 

*Many  writers,  such  as  Hayward  in  "Vector  Algebra  and  Trigonometry," 
and  Stringham  in  "  Uniplanar  Algebra,"  treat  this  product  of  coaxial  quater- 
nions as  if  it  were  the  product  of  vectors.  This  is  the  fundamental  error  in  the 
Argand  method. 


ADDITION    OF    VECTORS    IN    SPACE.  25 

Art.  5.    Addition  of  Vectors  in  Space. 

A  vector  in  space  can  be  expressed  in  terms  of  three  inde- 
pendent components,  and  when  these  form  a  rectangular  set 
the  directions  of  resolution  are  expressed  by  i,j,  k.  Any  vari- 
able vector  R  may  be  expressed  as  R  =  rp  =  xi+yj -\-zk,  and 
any  constant  vector  B  may  be  expressed  as 
B  =  bj3  =  b1z  +  bJ+b9k. 

In  space  the  symbol  p  for  the  direction  involves  two  ele- 
ments.     It  may  be  specified  as 

xi  -f-  yj '  +  zk 
P=  S+f  +  z*' 
where  the  three  squares  are  subject  to  the  condition  that  their 
sum  is  unity.  Or  it  may  be  specified  by  this  notation,  0//61, 
a  generalization  of  the  notation  for  a  plane.  The  additional 
angle  <p/  is  introduced  to  specify  the  plane  in  which  the  angle 
from  the  initial  line  lies. 

If  we  are  given  R  in  the  form  r<p//9,  then  we  deduce  the 
other  form  thus : 

R  =  r  cos  6  .  i  -\-  r  sin  6  cos  <p  .j  -\-  r  sin  8  sin  0  .  k. 
If  R  is  given  in  the  form  xi  -f-  yj '  -\-  2k,  we  deduce 


r=   4/V  +/  -f  z*  tan-1  -  /   tan-1 -f- 


For  example,    B  =  10  3Q°//45 

=  10  cos  450.  i-\-  10  sin  450  cos  300  ./■+  10  sin  450  sin  300 .  k. 

Again,  from  C  =  3? '+  4/+  5^  we  deduce 

..V41 


C=   V9  +  16+  25  tan"1"  /I  tan" 
=  7.07  51^47/64^9. 


To  find  the  resultant  of  any  number  of  component  vectors 
applied  at  a  common  point,  let  Rlf  Rq,  .  .  .  RH  represent  the  n 
vectors  or, 


26  VECTOR    ANALYSIS    AND    QUATERNIONS.. 


Rn  =  xj  +  ynj  -\-  znk  ; 
then  2R  =  {2x)i  +  (2y)f  +  (2z)k 


and  r  =  V(2x)2  +  {2y)*  +  (2z)\ 


2z       ,         n     V{2yy  +  (2zy 

tan0  =  -^     and     tan  d  =     v    JJ        — '-. 

Successive  Addition. — When  the  successive  vectors  do  not 
lie  in  one  plane,  the  several  elements  of  the  area  enclosed  will 
lie  in  different  planes,  but  these  add  by  vector  addition  into  a 
resultant  directed  area. 

Prob.  23.  Express  A  =  4/  —  5/  +  6k  and  B  =  52  +  6/  —  "jk  in 
the  form  r07/#.      (Ans.  8.8  i^7/63°  and  10.5  3~n7/6i0.5.) 

Prob.  24.  Express  C  =  123  57°//i42°  and  ^  =  456  65°//2oo° 
in  the  form  #z  -\- yj  -\-  zk. 

71    /  /7t  7t    /  /     7Z 

Prob.  2^.   Express  -c  =  100  —  //  -    and   F  =  1000  —  //  •?—   in 
°  4//  3  6//  ^4 

the  form  #2  -\-yj  +  2/£. 

Prob.  26.   Find  the  resultant  of  10  20°//3o°,  20  3o°//4o°,  and 

30  4^7/V5-  _ 

Prob.  27.  Express  in  the  form  r<p//6  the  resultant  vector  of 
it  -\-  2/  —  3k,  41  —  5/  +  6£,  and  —  7/  +  8/  +  9^. 


Art.  6.    Product  of  two  Vectors. 

Rules  of  Signs  for  Vectors  in  Space. — By  the  rules  i*  =  -{-, 
J"  =  +,  ij  =  k,  and  ji  =  —  k  we  obtained  (p.  432)  a  product  of 
two  vectors  containing  two  partial  products,  each  of  which  has 
the  highest  importance  in  mathematical  and  physical  analysis. 
Accordingly,  from  the  symmetry  of  space  we  assume  that  the 
following  rules  are  true  for  the  product  of  two  vectors  in  space  : 

*  =  +>  /"  =  +>  &  =  +> 

if  =  &>  Jk  =  h  ki  =  y, 

ji  =  —  k,  kj '  =  —  i,  ik  =  — j. 

The  square  combinations  give  results  which  are  indepen- 


PRODUCT  OF  TWO  VECTORS.  27 

dent  of  direction,  and  consequently  are  summed  by  simple 
addition.  The  area  vector  determined  by 
2  and/ can  be  represented  in  direction  by  k, 
because  k  is  in  tri-dimensional  space  the  axis 
which  is  complementary  to  z  and/.  We  also 
observe  that  the  three  rules  ij  —  k,  jk  =  i, 
ki  =j  are  derived  from  one  another  by  cyc- 
lical permutation ;  likewise  the  three  rules 
ji  =  —  k,  kj  =  —  i,  ik  =  — j.  The  figure  shows  that  these 
rules  are  made  to  represent  the  relation  of  the  advance  to  the 
rotation  in  the  right-handed  screw.  The  physical  meaning  of 
these  rules  is  made  clearer  by  an  application  to  the  dynamo  and 
the  electric  motor.  In  the  dynamo  three  principal  vectors  have 
to  be  considered  :  the  velocity  of  the  conductor  at  any  instant, 
the  intensity  of  magnetic  flux,  and  the  vector  of  electromotive 
force.  Frequently  all  that  is  demanded  is,  given  two  of  these 
directions  to  determine  the  third.  Suppose  that  the  direction 
of  the  velocity  is  i,  and  that  of  the  flux/,  then  the  direction  of 
the   electromotive  force  is  k.     The   formula   ij  =  k  becomes 

velocity  flux  =  electromotive-force, 
from  which  we  deduce 

flux  electromotive-force  =  velocity, 
and  electromotive-force  velocity  =  flux. 

The  corresponding  formula  for  the  electric  motor  is 
current  flux  =  mechanical-force, 
from  which  we  derive  by  cyclical  permutation 

flux  force  =  current,     and     force  current  =  flux. 

The  formula  velocity  flux  =  electromotive-force  is  much 
handier  than  any  thumb-and-finger  rule  ;  for  it  compares  the 
three  directions  directly  with  the  right-handed  screw. 

Example. — Suppose  that  the  conductor  is  normal  to  the 
plane  of  the  paper,  that  its  velocity  is  towards  the  bottom,  and 
that  the  magnetic  flux  is  towards  the  left ;  corresponding  to 
the  rotation  from  the  velocity  to  the  flux  in  the  right-handed 
•screw  we  have  advance  into  the  paper :  that  then  is  the  direc- 
tion of  the  electromotive  force. 

Again,  suppose  that  in  a  motor  the  direction  of  the  current 


28 


VECTOR    ANALYSIS    AND    QUATERNIONS. 


along  the  conductor  is  up  from  the  paper,  and  that  the  mag- 
netic flux  is  to  the  left ;  corresponding  to  current  flux  we  have 
advance  towards  the  bottom  of  the  page,  which  therefore  must 
be  the  direction  of  the  mechanical  force  which  is  applied  to 
the  conductor. 

Complete  Product  of  two  Vectors. — Let  A  ==  axi-\-aJ  -\-a%k 
and  B  =  bxi-\-bJ  -\-  b3k  be  any  two  vectors,  not  necessarily 
of  the  same  kind  physically,  Their  product,  according  to  the 
rules  (p.  444),  is 

AB  =  (aj  +  a  J  -f  a3k){bxi  -f-  bj  +  b,k), 
=  axbxii-\-  ajj^jj  -j-  a.bjzk, 

+  ajjjk  +  ajbtkj  -\-  atbtki  +  a  folk  +  axbjj  +  aj>ji 
=  axbx  +  a2b2  +  a3b3, 

+  (a,b3  -  aja)i+  [a3bx  —  axb3)j-\-  (axb2  —  a2bx)k 
=  axbx-{-aibi-{-a3b3-\~     ax     a2     a3 

bx      b2     b3 
i      j      k 

Thus   the   product  breaks  up   into    two   partial   products,, 
namely,  axbx-\-  aJ>%-\-  a3b3,  which  is  independent  of  direction,  and 
ax     a2     a3 

bx     b2     b3     ,  which  has  the  direction  normal  to  the  plane  of 
i     j      k 

A  and  B.  The  former  is  called  the  scalar  product,  and  the 
latter  the  vector  product.    ■' 

In  a  sum  of  vectors,  the  vectors  are  necessarily  homogene- 
ous, but  in  a  product  the  vectors  may  be  heterogeneous.  By 
making  a3  =  b3  =  o,  we  deduce  the  results  already  obtained 
for  a  plane. 

Scalar  Product  of  two  Vectors. — The  scalar  product  is  de- 
noted as  before  by  SAB.  Its  geometrical 
meaning  is  the  product  of  A  and  the  orthog- 
onal projection  of  B  upon  A.  Let  OP  rep- 
resent A,  and  OQ  represent  B,  and  let  OL, 
P  LM,  and  MN  be  the  orthogonal  projections 
upon  OP  of  the  coordinates  bxi,  b2j,  b3k  re. 
spectively.  Then  ON  is  the  orthogonal  pro- 
jection of  OQ  and 


PRODUCT  OF  TWO  VECTORS.  29 

OP  X  ON  =  OP  X  (OL  +  LM  +  MN), 

\    a  a  a  j 

=  axbx  -f-  « A  +  a*b3  —  SAB. 

Example.  —  Let  I  the  intensity  of  a  magnetic  flux  be 
B  =  bxi-\-  bJ-\-  b3k,  and  let  the  area  be  5=  sxi  -f-  sJ-\-  s%k  ; 
then  the  flux  through  the  area  is  SSB  =  bxsx  -\-  b%s^  -\-  b3s%. 

Corollary  i. — Hence  SB  A  =  SAB.     For 

bxax  +  d9a9  +  b3a3  =  axbx-\-  a,b,  -\-  a3b3 . 

The  product  of  B  and  the  orthogonal  projection  on  it  of  A 
is  equal  to  the  product  of  A  and  the  orthogonal  projection  on 
it  of  B.  The  product  is  positive  when  the  vector  and  the  pro- 
jection have  the  same  direction,  and  negative  when  they  have 
opposite  directions. 

Corollary  2. — Hence  A*  =  ax  ~\-a*-\-a3  =<£.  The  square  of 
A  must  be  positive  ;  for  the  two  factors  have  the  same  direction. 

Vector  Product  of  two  Vectors. — The  vector  product  as 
before  is  denoted  by  VAB.  It  means  the  product  of  A  and 
the  component  of  B  which  is  perpendicular  to  A,  and  is  rep- 
resented by  the  area  of  the  parallelogram  formed  by  A  and  B. 
The  orthogonal  projections  of  this  area  upon  the  planes  of  jk, 
ki,  and  if  represent  the  respective  components  of  the  product. 
For,  let  OP  and  OQ  (see  second  figure  of  Art.  3)  be  the  or- 
thogonal projections  of  A  and  B  on  the  plane  of  i  andj  ;  then 
the  triangle  OPQ  is  the  projection  of  half  of  the  parallelogram 
formed  hysA  and  B.  But  it  is  there  shown  that  the  area  of 
the  triangle  OPQ  is  \{axb^  —  ajbx).  Thus  (axb„  —anbx)k  denotes 
the  magnitude  and  direction  of  the  parallelogram  formed  by 
the  projections  of  A  and  B  on  the  plane  of  i  and/.  Similarly 
(a2b3  —  a3b2)i  denotes  in  magnitude  and  direction  the  projec- 
tion on  the  plane  of  j  and  k,  and  (a3bx  —  axb3)j  that  on  the 
plane  of  k  and  i. 

Corollary  1.— Hence  NBA  =  —  NAB. 

Example. — Given  two  lines  A  =  ji  —  io;'-(-  3^  and  B  = 
—  gi-\-4j  —  6k;  to  find  the  rectangular  projections  of  the  par- 
allelogram which  they  define  : 


30  VECTOR    ANALYSIS    AND    QUATERNIONS. 

VAB  =  (60  -  12)/  +  (-  27  -f  42)7  +  (28  -  go)£ 
=  48/  -f-  1 5/  —  62^. 

Corollary  2. — If  y2  is  expressed  as  aa  and  B  as  £/?,  then 
S^.5  =  ab  cos  «/?  and  VAB  =  ab  sin  or/?  .  aft,  where  a/?  de- 
notes the  direction  which  is  normal  to  both  a.  and  fi,  and 
drawn  in  the  sense  given  by  the  right-handed  screw. 

Example.— Given  A  =  rcp//d  and  B  =  r'^Tf/d'.     Then 

SAB  =  rr'  cos  ^//J_^//0^ 

=  r/|cos  0  cos  0'  -f-  sin  6  sin  0'  cos  (<p'  —  0)}. 

Product  of  two  Sums -of  non-successive  Vectors. — Let  A  and 
i?  be  two  component  vectors,  giving  the  resultant  A  -\-  B,  and 
let  C  denote  any  other  vector  having  the  same  point  of  appli- 
cation. 

Let  A  =  axi  -\-  a  J  -f-  azk, 

-A+B 

B  =  bj+bJ+b;k, 


C  =  c,i-\-  cj  +  c3k. 
Since  A  and  B  are  independent  of  order, 
A  +  B  =  {a,  +  6t)i  +  (A,  +  £,)./  +  fa  +  W 
consequently  by  the  principle  already  established 

S(^  +  B)C  =  fa  +  £>,  +  fa  +  *,K  +  (a,  +  ^V, 

=  S^c7+S£C. 
Similarly  V(i4  +  £)<7  =  {fa  +  J,y,  -  fa  +  *X}i  +  etc. 

=   fa^    —   *,*,)*    +    ( ^8    —   KC&  +  •   •   • 

=  V^<r  +  V^(T. 

Hence  (A  +  B)C  =  AC+ BC. 

In  the  same  way  it  may  be  shown  that  if  the  second  factor 
consists  of  two  components,  C  and  D,  which  are  non-successive 
in  their  nature,  then 

(A  +  B){C  +  D)  =  AC+  AD  +  BC  +  BD. 


PRODUCT  OF  THREE  VECTORS.  31 

When  A  -j-  B  is  a  sum  of  component  vectors 
(A  +  BY  =  A*  +  B2  +  AB  -\-  BA 
=  A2  +  B>  +  2SAB. 

Prob.  28.  The  relative  velocity  of  a  conductor  is  S.W.,  and  the 
magnetic  flux  is  N.W.;  what  is  the  direction  of  the  electromotive 
force  in  the  conductor  ? 

Prob.  29.  The  direction  of  the  current  is  vertically  downward, 
that  of  the  magnetic  flux  is  West;  find  the  direction  of  the  mechani- 
cal force  on  the  conductor. 

Prob.  30.  A  body  to  which  a  force  of  2/  +  3/  +  4k  pounds  is 
applied  moves  with  a  velocity  of  5/+  6/+  7k  feet  per  second;  find 
the  rate  at  which  work  is  done. 

Prob.  31.  A  conductor  8/+  9/  +  \ok  inches  long  is  subject  to 
an  electromotive  force  of  11/+  12/+  13^  volts  per  inch;  find  the 
difference  of  potential  at  the  ends.  (Ans.  326  volts.) 

Prob.  32.  Find  the  rectangular  projections  of  the  area  of  the 
parallelogram  defined  by  the  vectors  A  =  12/—  23/'—  34^  and 
■B  =  -45/-  567  +  67^. 

Prob.  33.  Show  that  the  moment  of  the  velocity  of  a  body  with 
respect  to  a  point  is  equal  to  the  sum  of  the  moments  of  its  com- 
ponent velocities  with  respect  to  the  same  point. 

Prob.  34.  The  arm  is  9/+  ii/'-f-  13k  feet,  and  the  force  applied 
at  either  end  is  17/  +  19/  +  23^  pounds  weight;  find  the  torque. 

Prob.  35.  A  body  of  1000  pounds  mass  has  linear  velocities  of  50 
feet  per  second  30° //450,  and  60  feet  per  second  6o0//22°.5;  find 
its  kinetic  energy. 

Prob.  36.  Show  that  if  a  system  of  area-vectors  can  be  repre- 
sented by  the  faces  of  a  polyhedron,  their  resultant  vanishes. 

Prob.  37.  Show  that  work  done  by  the  resultant  velocity  is  equal 
to  the  sum  of  the  works  done  by  its  components. 

Art.  7.    Product  of  Three  Vectors. 

Complete  Product. — Let  us  take  A  =  aj  -f-  a  J  -\-  a3k, 
B  =  bxi  -f  bj  -\~  b%k,  and  C  =  c\i  +  cJ-\-  c^k.  By  the  product 
of  A,  B,  and  C  is  meant  the  product  of  the  product  of  A  and 
B  with  C,  according  to  the  rules    p.  444).     Hence 

ABC  =  (aj),  +  a,b2  +  a3b3)(cj  -f  cj  +  c,k) 

-HO* A  -  "A)i+  (#A  -  aA)j'+  ("A  ~  *J>W\(cxi-\-cJ+  ctk) 
=  (a  A  +  a  A  +  a%b^{cj  +  cj  -\-  c%k)  ( 1) 


82 


VECTOR    ANALYSIS    AND    QUATERNIONS. 


+ 


a2  a% 

a%  a1 

ax  a, 

K  K 

K  K 

K  K 

Ci                c*               ^ 

i 

J 

k 

(2)  + 


ax 

a,  a3 

K 

Kb, 

c, 

c*  c% 

(3) 


Example. — Let  A  =  li  -f-  2j-\-  $k,  B  =  42  -f-  5/  -|-  6£,  and 
C=  71 +  8J+ 9&.     Then 

(i)=(4+io+i8)(7^  +  8/+9^)=32(72  +  8y+9^). 


(2)  = 

-3 

6 

-3| 

7 

8 

9 

i 

J 

k\ 

(3)  = 

1     2 

3 

—  0. 

4     5 

6 

7    8 

9 

If  we  write  A  =  ««,  B  =  b/3,  C  =  cy,  then 

ABC  =  abc  cos  a/3  .  y  (1) 

-j-  abc  sin  «•/?  sin  afiy .  a/?;/  (2) 

-f-  abc  sin  a/3  cos  ar/3;/,  (3) 

where  cos  a/3y  denotes  the  cosine  of  the  angle  between  the 

directions  a/3  and  y,  and  afiy  denotes  the  direction  which  is 

normal  to  both  a/3  and  y. 

We  may  also  write 

ABC  =  SAB.  C+V(VAB)C+  S{VAB)C. 

(1)  (2)  (3) 

First  Partial  Product. — It  is  merely  the  third  vector  multi- 
plied by  the  scalar  product  of  the  other  two,  or  weighted  by 
that  product  as  an  ordinary  algebraic  quantity.  If  the  direc- 
tions are  kept  constant,  each  of  the  three  partial  products  is 
proportional  to  each  of  the  three  magnitudes. 

Second  Partial  Product. —  The  second  partial  product  may 
be  expressed  as  the  difference  of  two  products  similar  to  the 
first.     For 

V(VAB) C  =  {-  ( V,  +  <VsK  +  (',*,  +  'A)*, } • 

+1  —  (V.  +  V.K  +  0v*a  +  ciaMJ 
+  {-  OV,  +  hc*)*s  +  Mi  +  c^b%\k. 


PRODUCT  OF  THREE  VECTORS. 


33 


By  adding  to  the  first  of  these  components  the  null  term 
(b1cJa1  —  cxafi^)i  we  get  —  SBC .  aj  +  SCA  .  bj,  and  by  treating 
the  other  two  components  similarly  and  adding  the  results  we 
obtain 

V{VAB)C  =  -  SBC .  A  +  SCA  .  B. 

The  principle  here  proved  is  of  great  use  in  solving  equa- 
tions (see  p.  455). 

Example. — Take  the  same  three  vectors  as  in  the  preced- 
ing example.     Then 

V(V;lff)C=-(28  +  40+54)(i*'  +  2/+3*) 

+  (7  +  i6  +  27)(4z-+5/  +  6£) 
=  78/  +  6/  —  66k. 
The  determinant  expression  for  this  partial  product  may 
also  be  written  in  the  form 


b,  K 


i    J 


\ba  K 


j  k 


+ 


a,  a, 

K  b, 


k    i 


It  follows  that  the  frequently  occurring  determinant  expression 


a,  a, 

b,  K 


d,  d„ 


+ 


a2  as 
b„  b. 


dnd. 


+ 


a3  ar 
K  b. 


dvd, 


means  S(VAB)(VCD). 

Third  Partial  Product. — From  the  determinant  expression 
for  the  third  product,  we  know  that 

S(VAB)C=S(VBC)A  =  S(VCA)B 
=  -  S{VBA)C  =  -  S(VCB)A  =  -  S{VAC)B. 
Hence  any  of  the  three  former  may  be  expressed   by  SABC, 
and  any  of  the  three  latter  by  —  SABC. 

The  third  product  S(VAB)C  is  represented  by  the  vol- 
ume of  the  parallelepiped  formed  by  the  vectors  A,  B,  C 
taken  in  that  order.  The  line  *VAB  v,ab 
represents  in  magnitude  and  direction 
the  area  formed  by  A  and  B,  and  the 
product  of  NAB  with  the  projection 
of  C  upon  it  is  the  measure  of  the 
volume  in  magnitude  and  sign.  Hence  the  volume  formed 
by  the  three  vectors  has  no  direction  in  space,  but  it  is  posi- 
tive or  negative  according  to  the  cyclical  order  of  the  vectors. 


34  VECTOR    ANALYSIS    AND    QUATERNIONS. 

In  the  expression  abc  sin  a/3  cos  a/3y  it  is  evident  that  sin  a/3 
corresponds  to  sin  0,  and  cos  a/3y  to  cos  0,  in  the  usual  for- 
mula for  the  volume  of  a  parallelepiped. 

Example. — Let  the  velocity  of  a  straight  wire  parallel  to 
itself  be  V  =  1000/300  centimeters  per  second,  let  the  intensity 
of  the  magnetic  flux  be  B  =  6000  /900  lines  per  square  cen- 
timeter, and  let  the  straight  wire  L  —  15  centimeters  600/  /450. 
Then  V  VB  =  6000000  sin  6o°  900/  /900  lines  per  centimeter  per 
second.  Hence  S{VVB)L  =15  X  6000000  sin  6o°  cos  0  lines 
per  second  where  cos0  =  sin  450  sin  6o°. 

Sum  of  the  Partial  Vector  Products. — By  adding  the  first 
and  second  partial  products  we  obtain  the  total  vector  product 
of  ABC,  which  is  denoted  by  V(ABC).  By  decomposing  the 
second  product  we  obtain 

V{ABC)  =  SAB.  C  -  SBC .  A  +  SCA  .  B. 
By  removing  the  common  multiplier  abc,  we  get 

V(aj3y)  =  cos  af3  .  y  —  cos  /3y .  a  -\~  cos  ya  .  /3. 
Similarly  V(/3ya)  —  cos  (3y  .  a  —  cos  ya .  /3  -f-  cos  a/3  .  y 
and  V(ya/3)  =  cos  ya  .  /3  —  cos  aj3  .  y  -f-  cos  j3y  .  a. 

These   three  vectors   have  the    same    magnitude,  for  the 
square  of  each  is 
cos2  a/3  -J-  cos2  /3y  -f  cos2  ya  —  2  cos  a/3  cos  /3y  cos  ya, 
that  is,  1  -{S(a/3y)\\ 

They  have  the  directions  respectively  of  a', 
(3r,  y' ,  which  are  the  corners  of  the  triangle 
whose  sides  are  bisected  by  the  corners  a, 
(3,  y  of  the  given  triangle. 

Prob.  38.  Find  the  second  partial  product  of 
g  2o°//30°,  10  3o°//40°,  n  45°//45°.  Also  the  third  partial 
product. 

Prob.  39.  Find  the  cosine  of  the  angle  between  the  plane  of 
/,/+#/,/+  nxk  and  /„/  +  m2j-\-nak  and  the  plane  of  lJ-\-m3j-\-nJt 
and  I J  +  mj  -\-  nfi. 

Prob.  40.  Find  the  volume  of  the  parallelepiped  determined  by 
the  vectors  ioo/+5q/'  +  25^,  50/4- i°/  +  8o/£,  and  —  75/+  407  —  8o£. 


COMPOSITION    OF    QUANTITIES.  35 

Prob.  41.  Find  the  volume  of  the  tetrahedron  determined  by  the 
extremities  of  the  following  vectors  :  3/  —  2/  -f-  \k,  —  4/  +  5/'  —  7>£, 
3/  —  y  —  2k,  8/  +  4/"  —  3& 

Prob.  42.  Find  the  voltage  at  the  terminals  of  a  conductor  when 
its  velocity  is  1500  centimeters  per  second,  the  intensity  of  the  mag- 
netic flux  is  7000  lines  per  square  centimeter,  and  the  length  of  the 
conductor  is  20  centimeters,  the  angle  between  the  first  and  second 
being  300,  and  that  between  the  plane  of  the  first  two  and  the  direc- 
tion of  the  third  6o°.  (Ans-  .91  volts.) 

Probr  43.  Let  a  =  2^7/10°,  0  =  307/25°,  V  =  4^7/35 °-  Find 
Vafiy,  and  deduce  Vftya  and  Vyafi. 


Art.  8.    Composition  of  Quantities. 

A  number  of  homogeneous  quantities  are  simultaneously- 
located  at  different  points ;  it  is  required  to  find  how  to  add  or 
compound  them. 

Addition  of  $.  Located  Scalar  Quantity. — Let  mA  denote  a 
mass  m  situated  at  the  extremity  of  the  radius- 
vector  A.     A  mass  m  —  m  may  be  introduced 
at  the  extremity  of  any  radius-vector  R,  so 

that 

mA  =  (m  —  m)R  -(-  mA 

=  mR  -f-  mA  —  mR 
=  mR  +  m(A  —  R). 
Here  A  —  R  is  a  simultaneous  sum,  and  denotes  the  radius- 
vector  from  the  extremity  of  R  to  the  extremity  of  A.  The 
product  m{A  —  R)  is  what  Clerk  Maxwell  called  a  mass- vector, 
and  means  the  directed  moment  of  m  with  respect  to  the  ex- 
tremity of  R.  The  equation  states  that  the  mass  m  at  the 
extremity  of  the  vector  A  is  equivalent  to  the  equal  mass  at 
the  extremity  of  R,  together  with  the  said  mass-vector  applied 
at  the  extremity  of  R.  The  equation  expresses  a  physical  or 
mechanical  principle. 

Hence  for  any  number  of  masses,  ml  at  the  extremity  of  Al9 
m7  at  the  extremity  of  A^,  etc., 

2mA  =  2mx  +  2{m(A  —  R)\, 


36  VECTOR    ANALYSIS    AND    QUATERNIONS. 

where  the  latter  term   denotes  the    sum  of  the  mass-vectors 
treated  as  simultaneous  vectors  applied  at  a  common  point. 

Since  2\m(A  —  R)}  =  2mA  —  2mR 

=  2mA  —  R2m, 

the  resultant  moment  will  vanish  if 

R  =  — tt; — ,       or      R2m  =  2mA 
2  m 

Corollary. — Let       R  =  xi  -j-  yj  -\-  zk, 
and  A  =aj  -j-  bj  -f-  £,£ ; 

then  the  above  condition  may  be  written  as 

xi  -\-  yj  -\-  zk  =■ ^ 

1   ■'•'    '  2m 

_  2 (ma)  .  i  .   (2mb)  .j  .   2(mc) .  k 
2m  2m  2m     ' 

2  (in  a)  2(mb)  2mc 

therefore  x=—^ — ,     y  =  — ^ — ,     z=  — — 

2m  2m  2m 


Example. — Given    5    pounds   at    10  feet   45°//30°  and    8 
pounds  at  7  feet  6o°//45°  !  find  the  moment  when  both  masses 


are  transferred  to  12  feet  75°//6o°. 

m1A1  =  5o(cos  30°?'  -f-  sin  300  cos  45°/+  sm  3°°  sin  45°^)» 
m2A^  =  56(cos  45°?-f-  sin  450  cos  6o°j-\-  sin  450  sin  60° k), 
(m1  -f-  m^)R  =  i56(cos  6o°z -f-  sin  6o°  cos  75^' -f-  sin  6o°  sin  75°/£), 
moment  =  mlA1  -j-  m2A^  —  (m1  -j-  w2)-^. 

Composition   of  a  Located  Vector  Quantity. — Let  FA  de- 
note a  force  applied  at  the  extremity  of  the  radius-vector  A. 
As  a  force  F —  F  may  introduced  at  the  ex- 
tremity  of  any  radius-vector  R,  we  have 

FA-(F-F)R  +  FA 
=  FR  +  V(A  -  R)F 

This  equation  asserts  that  a  force  F  applied 
at  the  extremity  of  A  is  equivalent  to  an  equal  force  applied 
at  the  extremity  of  R  together  with  a  couple  whose  magnitude 


COMPOSITION    OF    QUANTITIES.  37 

and  direction  are  given  by  the  vector  product  of  the  radius- 
vector  from  the  extremity  of  R  to  the  extremity  of  A  and  the 
force. 

Hence  for  a  system  of  forces  applied  at  different  points, 
such  as  Fx  at  Ax,  Fa  at  A2,  etc.,  we  obtain 

2(FA)  =  2(FX)  +  2V(A  -  R)F 
=  (2F)R  +  2V(A-R)F. 

Since  2V(A  -  R)F  =  2VAF—  2VRF 

=  2VAF-VR2F 
the  condition  for  no  resultant  couple  is 

VR2F=2VAF, 

which  requires  2F  to  be  normal  to  2VAF. 

Example. — Given  a  force  it  4-  2/ -f-  3^  pounds  weight  at 
A1 +  5J-{-6k  Ieet>  an^  a  force  of  ji  -\-  gj  -f-  1 1  k  pounds  weight 
at  \oi -\-  12/  -f-  \\k  feet;  find  the  torque  which  must  be  sup- 
plied when  both  are  transferred  to  2z  -J-  5/  -\-  3k,  so  that  the 
effect  may  be  the  same  as  before. 

VAxFx  =  v-6/  +  3k, 
VAtF9  =  6i-  I2/4-6&, 
2VAF=gi-  iS/4-gk, 
2F=  Si  4-  11/  +  14/&, 
VR2F=37t-  4j—  i8£, 
Torque  =  —  28/—  14 j  4-  27k. 
By  taking  the  vector  product  of  the  above  equal  vectors 
with  the  reciprocal  of  .2.F  we  obtain 

V{(V^^}=V{(2V^/^|. 

By  the  principle  previously  established  the   left    member 

resolves  into  —  R  -j-  SR^rp.  2F;   and   the  right    member   is 

equivalent  to  the  complete  product  on  account  of  the  two 
factors  being  normal  to  one  another;  hence 

-R4-  SR^  .  2F  =  2(VAF)~; 


38  VECTOR    ANALYSIS   AND    QUATERNIONS. 

that  is,  R  =  ~2(VAF)  +  SR^?  •  2F. 

(I)  (2) 

The  extremity  of  R  lies  on  a  straight  line  whose  perpen- 
dicular is  the  vector  (i)  and  whose  direction  is  that 
of  the  resultant  force.  The  term  (2)  means  the 
projection  of  R  upon  that  line. 

The  condition  for  the  central  axis  is  that  the 
resultant    force    and    the   resultant    couple   should 
have  the  same  direction ;  hence  it  is  given  by 
V  { 2VAF  -  VR2F\  2F=o; 
that  is,  V{VR2F)2F  =  V(2AF)2F. 

By  expanding  the  left  member  according  to  the  same  prin- 
ciple as  above,  we  obtain 

—  (2F)2R  4-  SR2F.  2F  =  V{2AF)2F; 

t  s  r  y  F 

therefore  R  =  ^p^2F(V2AF)  +  -^w .  2F 


=  v{iTyV2AF)  +  SR^F.2F. 


This  is  the  same  straight  line  as  before,  only  no  relation  is 
now  imposed  on  the  directions  of  2F  and  2VAF;  hence  there 
always  is  a  central  axis. 

Example. — Find  the  central  axis  for  the  system  of  forces 
in  the  previous  example.  Since  2  F=  8z'-f-  11/ -j-  14A  the 
direction  of  the  line  is 

V64  -j-  121  +  196' 

Since  ~  =  8'  +  1 1/  +  14*  and  ^VAF  =  9i  -  l8/  +  9£,  the 
2,-r  3bI 

perpendicular  to  the  line  is 

Prob.  44.  Find  the  moment  at  cioVMo0  of  IO  pounds  at  4  feet 
io°//2o°  and  20  pounds  at  5  feet  3o0//i20°. 


SPHERICAL    TRIGONOMETRY. 


>0 


Prob.  45.  Find  the  torque  for  4/ +37+  2k  pounds  weight  at 
2/  —  3/'  +  i£  feet,  and  2/  —  1/  —  i>£  pounds  weight  at  —  3/  +  4/'  +  5^ 
feet  when  transferred  to  —  32  +  2/  —  4^  feet. 

Prob.  46.  Find  the  central  axis  in  the  above  case. 

Prob.  47.  Prove  that  the  mass-vector  drawn  from  any  origin  to  a 
mass  equal  to  that  of  the  whole  system  placed  at  the  center  of  mass 
of  the  system  is  equal  to  the  sum  of  the  mass-vectors  drawn  from 
the  same  origin  to  all  the  particles  of  the  system. 


Art.  9.     Spherical  Trigonometry. 

Let  i,j,  k  denote  three  mutually  perpendicular  axes.  In 
order  to  distinguish  clearly  between  an  axis  and  a  quadrantal 
version  round  it,  let  i     ,j     ,  kT  2  denote  k 

quadrantal  versions  in  the  positive  sense 
about   the   axes  i,J,  k  respectively.     The 
directions  of  positive  version  are  indicated    ~j\ 
by  the  arrows. 

By  2rr'S    2  is  meant  the  product  of  two 
quadrantal  versions  round   2";  it  is  equiv-  -k 

alent  to  a  semicircular  version  round  i\  hence  ir/Hv/'1  =  1"  =  — . 
Similarly/r/2/r/2  means  the  product  of  two  quadrantal  versions 
round/,  and/V/a  =f  =  — •     Similarly  kw/*kn/*  =  F  =  -. 

By  z"r/yr/2  is  meant  a  quadrant  round  i  followed  by  a  quad- 
rant round/;  it  is  equivalent  to  the  quadrant  from/  to  2,  that 
is,  fro  —  kn/\  Butjn/*i7r/i  is  equivalent  to  the  quadrant  from  —  2 
to  — /,  that  is,  to  kn/\  Similarly  for  the  other  two  pairs  of 
products.     Hence  we  obtain  the  following 

Rules  for  Versors. 


7'7r/2    -7r/2 


-Va  :n/i 


Va  Wa 


j«/ij*/i  _  k    *&"      =  — 


' "",  '2 


fug!*  _  _  ,-A         k^f'9  =  i 


7  7r/3^7r/3  ^.jt/j 


l"*k~U 


;w/a 


-2        =  —  j      ,  „      „ 

The  meaning  of  these  rules  will  be  seen  from  the  follow 
ing  application.      Lei    li  +  tnj  +  nk  denote    any  axis,  then 


40  VECTOR    ANALYSIS    AND    QUATERNIONS. 

•(/*'  +  mj  -\-  nk)*/*  denotes  a  quadrant  of  angle  round  that  axis. 
This  quadrantal  version  can  be  decomposed  into  the  three 
rectangular  components  lin/i,  mjn'*,  nk1 r/" ;  and  these  components 
are  not  successive  versions,  but  the  parts  of  one  version.  Sim- 
ilarly any  other  quadrantal  version  (I'i  -f-  m'j  -j-  n'kf'*  can  be 
resolved  into  l'in/i,  m'j"''*,  n'kn/*.  By  applying  the  above  rules, 
we  obtain 
(/«  +  mj  +  nk)"'\l'i  +  m'j  +  rik?u 

=  (#*/«  4-  mff\  +  nkw/%){l'?/%  +  m'f/%  +  tf'F78) 

=  —  (11 '  -\-  mm'  -\-  nn') 

-  (mri  -  m'nY1"  -  (»/'  -  «,/)//a  -  (/»«'  -  l'm)kn/* 

=  —  (//'  -f-  #z;tz'  -|~  «»') 

_  |(wy  _  «'»),' 4-  («/'  -  n'l)j-\-(lm!  -  /'m)k\n/\ 

Product  of  Two  Spherical  Versors. — Let  ft  denote  the  axis 
and  b  the  ratio  of  the  spherical  versor  PA,  then  the  versor 
itself  is  expressed  by  fih.  Similarly  let  y 
denote  the  axis  and  c  the  ratio  of  the 
spherical  versor  AQ,  then  the  versor  itself 
is  expressed  by  yc. 

Now     fib  =cosb  +  sin  b .  ff'\ 
and  yc  =  cos  c  -f-  sin  c  .  yn/2 ; 

therefore 
J3y  =  (cos  b  -f  sin  b  .  /3w/a)(cos  c  +  sin  c  .  yw/a) 

—  cos  b  cos  £  -j-  cos  £  sin  £  .  y^*  -f-  cos  £  sin  £ .  /3,r/* 

+  sin  bsinc.  ^^y'\ 
But  from  the  preceding  paragraph 

fl<l*y*l*    _    _    COS  yffj,   _    sin  fiy   .   j^»/«  . 

therefore         fibyc  =  cos  #  cos  c  —  sin  £  sin  c  cos  /?;/  (i) 

-|-  -jcos^sin^.  y-f-  cos  £  sin  b .  ft  —  sin  £  sin  <:sin  /?;/  . /^f^2.  (2) 
The  first  term  gives  the  cosine  of  the  product  versor  ;  it  is 
equivalent  to  the  fundamental  theorem  of  spherical  trigonom- 
etry, namely, 

cos  a  =  cos  b  cos  c  -f-  sin  b  sin  c  cos  A, 


SPHERICAL    TRIGONOMETRY.  41 

where  A   denotes  the  external  angle  instead  of  the  angle  in- 
cluded by  the  sides. 

The  second  term  is  the  directed  sine  of  the  angle ;  for  the 
square  of  (2)  is  equal  to  1  minus  the  square  of  (1),  and  its  di- 
rection is  normal  to  the  plane  of  the  product  angle.* 


Example.— Let  J3  =  307/45 °  and  y  =  607/300.     Then 
cos  fiy  =  cos  450  cos  300  -4-  sin  450  sin  300  cos  300, 
and  sin  fiy  .  fiy  =  Vfiy ; 

but  J3  =  cos  450 i  -f-  sin  450  cos  30°/+ sin  450  sin  30°^, 
and  y  =  cos  300  i  -f-  sin  300  cos  6o°j-\-  sin  300  sin  6o°  k  ; 
therefore 

Vfiy=  I  sin  450  cos30°sin30°sin6o° 

—  sin  450  sin  300  sin  300  cos6o°}z' 
-f-  |sin45°sin  300  COS300  —  COS450  sin  300  sin6o°}/ 
-J-  { cos  450  sin  300  cos  6o°  —  sin  450  cos  300  cos  300  }/£. 

Quotient  of  Two  Spherical  Versors. — The  reciprocal  of  a 
given  versor  is    derived    by  changing  the  sign  of  the  index ; 
y~c   is   the    reciprocal  of  y\     As  /?*  =  cos  b  -f-  sin  b .  /3w/a,  and 
y~c  =  cos  £  —  sin  c  .y"  ', 
fiby~c  =  cos  £  cos  £  -]-  sin  b  sin  £  cos  /?j/ 

+  jcos  c  sin  b .  /3—  cosb  sin  c.y  -\~  sin  <5  sin  £  sin  fiy .  fiy  j-"^9* 

Product  of  Three  Spherical  Versors. — Let 

aa  denote  the  versor  PQ,  fib  the  versor  QR, 

and  yc  the  versor  RS ;  then  aaj3byc  denotes 

PS.     Now  «7?V  P 

Q 

=  (cos  a  -j-  sin  a .  «ff/9)(cos  $  -f-  sin  b .  /f/2)(cos  <:  -f-  sin  c .  y"1'*) 

—  cos  a  cos  3  cos  <:  (1) 

+  cos  a  cos  b  sin  £.  ^'r/a  -{-  cos  a  cos  r  sin  b .  /J"79 

-j-  cos  b  cos  £  sin  «  .  a  (2) 

-f-  cos  a  sin  £  sin  <: .  {3n/iyn/i  -j-  cos  £  sin  «  sin  c .  ar/ayr/i 

-f-  cos  <:  sin  «  sin  ^  .  a"^'     (3) 
*  Principles  of  Elliptic  and  Hyperbolic  Analysis,  p.  2. 


42  VECTOR    ANALYSIS    AND    QUATERNIONS. 

-J-  sin  a  sin  b  sin  c ,  an/*ft"   y     .  (4)* 

The  versors  in  (3)  are  expanded  by  the  rule  already  ob- 
tained, namely, 

fTl*Y*U  _  _  cos  py  _  sin  py .  j^/A 

The  versor  of  the  fourth  term  is 

a"/*p«/y/>  =  _  (cos  ap  +  sin  a/S .  ~^p"/*)y*/* 

=  —  cos  aft  .  yt/i-\-sm  aft  cos  afty-\-sin  a/3  sin  a  fty .  aftyn'2. 


Now  sin  a/3  sin  a:/5}/ .  <*/?}/  =  cos  ay  .  ft  —  cos  fty  .  a  (p.  45 1), 
hence  the  last  term  of  the  product,  when  expanded,  is 

sin  a  sin  b  sin  c\  —  cos  aft .  y"U  +  cos  ay  .  ft"/* 

—  cos  fty  .  an/*  -f-  cos  afty\. 
Hence 
cos  aaftbyc  =  cos  #  cos  b  cos  £  —  cos  a  sin  £  sin  c  cos  /?;/ 

—  cos  b  sin  #  sin  £  cos  ay  —  cos  c  sin  a  sin  <5  cos  aft 
-f-  sin  #  sin  b  sin  £  sin  «/3  cos  afty, 

and,  letting  Sin  denote  the  directed  sine, 

Sin  aaftbyc  =  cos  a  cos  b  sin  c .y  -\-  cos  «  cos  c  sin  b  .  ft 

-f-  cos  #  cos  £  sin  «  .  «  —  cos  a  sin  £  sin  c  sin  /?;/  .  /?;/ 

—  cos  b  sin  «  sin  c  sin  #;/ .  ay 

—  cos  £  sin  a  sin  #  sin  or/?  .  aft 

—  sin  a  sin  b  sinc\zosaft  .  y  —  cos  a?^  . /5-j-cos /5/ .  or}.*' 

Extension  of  the  Exponential  Theorem  to  Spherical  Trigo- 
nometry.— It  has  been  shown  (p.  458)  that 

cos  ftbyc  =  cos  b  cos  c  —  sin  b  sin  c  cos  fty 
and 

(sin  ftbyc)n/*  =  cos  c  sin  £  .  /f /f  +  cos  b  sin  c.  yn/2 

—  sin  b  sin  c  sin  fty .  fty"   . 

Now  cos  b  =  1 r  -4-  —r  —  2t  +  etc. 

2  !       4  !        6  ! 

*  In  the  above  case  the  three  axes  of  the  successive  angles  are  not  perfectly- 
independent,  for  the  third  angle  must  begin  where  the  second  leaves  off.  But 
the  theorem  remains  true  when  the  axes  are  independent  ;  the  factors  are  then- 
quaternions  in  the  most  general  sense. 


SPHERICAL    TRIGONOMETRY.  43 

b3        bb 
and  sin  b  =  b  —  —r  +  — r  —  etc. 

Substitute  these  series  for  cos  b,  sin  b,  cos  c,  and  sin  c  in 
the  above  equations,  multiply  out,  and  group  the  homogeneous 
terms  together.     It  will  be  found  that 

cos  (3byc  =  1  —  —{b2  -f  2bc  cos  /3y  +  c*\ 

+  -\{ bk  +  4b3 c  cos  /3r  +  6£V  +  4bc5  cos  /?r  +  c*\ 

-  ~{be  +  6£V  cos  /?x  +  I5*V  +  2obV  cos  /?r 

+  is^V  +  6bc&  cos  /?r  +  c<\  +  .  .  ., 
where  the  coefficients  are  those  of  the  binomial  theorem,  the 
only  difference  being  that  cos  /3y  occurs  in  all  the  odd  terms 
as  a  factor.  Similarly,  by  expanding  the  terms  of  the  sine,  we 
■obtain 

<Sin  fiff*  =  b.ff/2  +  c.  y»/z  -  be  sin  f3y .  fiyn/2 

-jj{b\f/*+sb*C.y"/*  +  lbc>.F<*  +  <».y^) 

+  -\{  be3  +  b*c\  sin  0y  .  jfyn/2 

+  ~-{b\  f?'2  +  $b*c .  y/2  +  lobY  .  /f /a 

+  lObV  •  y"/z  +  $bc* .  /T/2  +  ^5 .  rff/2 } 

~  ^T  |  ^  +  ^V  +  ^  }  Knfiy.fiP"*-.  .  . 

By  adding  these  two  expansions  together  we  get  the  ex« 
pansion  for  (3hyc,  namely, 

/36yc=i-\-b.f/i  +  c.yn/2 

-  -\  !  V  +  2^(cos  /Jy  -f  sin  /3y  .  Jy*'2)  +  <:'  f 

-f  -^  £4  -f-  4*V(cos  /?;/  +  sin  /3y  .  /^//2)  +  6£V 

+  4^3(cos  /?r  +  sin  fty  .  Jy"/2)  -!_«*}  +  ,.. 


44  VECTOR    ANALYSIS    AND    QUATERNIONS. 

By  restoring  the  minus,  we  find  that    the  terms   on    the 
second  line  can  be  thrown  into  the  form 

and  this  is  equal  to 

where  we  have  the  square  of  a  sum  of  successive  terms.     In  a 
similar  manner  the  terms  on  the  third  line  can  be  restored  to 
p .  fi*l*  _j_  $b*c .  § yw/*  +  3bc* .  /3n/2rn  -f  c" .  y*{"/2\ 

that  is,  ±.{3.^  +  c.r^\\ 

Hence 

+  j[\b.^+c.y"/r  +  ±\b.f/*  +  c.y^\*  + 

Extension  of   the    Binomial  Theorem. — We   have   proved 

above    that  //^V^2  =  ^/2  +  ^/2  provided   that  the    powers 

of  the  binomial  are  expanded  as  due  to  a  successive  sum,  that 

is,  the  order  of  the  terms  in  the  binomial  must  be  preserved. 

Hence  the  expansion  for  a  power  of  a  successive  binomial  is 

given  by 

|0  .  /T72  +  c  .  yl%\"  =  bn  .  P"n/2  +  nbn~\c  .  $*-***/*  y*l* 

n(n—  i) 
+ '-Ir-V./fr-'WDyv+etc. 

*  At  page  386  of  his  Elements  of  Quaternions,  Hamilton  says:  "In  the 
present  theory  of  diplanar  quaternions  we  cannot  expect  to  find  that  the  sum  of 
the  logarithms  of  any  two  proposed  factors  shall  be  generally  equal  to  the 
logarithm  of  the  product  ;  but  for  the  simpler  and  earlier  case  of  coplanar 
quaternions,  that  algebraic  property  may  be  considered  to  exist,  with  due 
modification  for  multiplicity  of  value."  He  was  led  to  this  view  by  not  dis- 
tinguishing between  vectors  and  quadrantal  quaternions  and  between  simul- 
taneous and  successive  addition.  The  above  demonstration  was  first  given  in 
my  paper  on  "  The  Fundamental  Theorems  of  Analysis  generalized  for  Space." 
It  forms  the  key  to  the  higher  development  of  space  analysis. 


COMPOSITION    OF    ROTATIONS. 


45 


Example.— Let  b^=  -^  and  c=\,  ft  =  30°/M£>  Y  =  6o0//30°. 
{b .  ^  -J-  c .  rn/2y  =  -5^  +  ^-1-  2^  cos  (3y  -4-  2^(sin  /S^)"78} 

=  ~(-ih>+-h  + 1\  cos  /?r)  -  &(sin  ^)'r/a. 
Substitute  the  calculated  values  of  cos  fiy  and  sin  /3y  (page  459). 
Prob.  48.  Find  the   equivalent  of   a  quadrantal  version   round 

V~z  1  1 

-j  -\ -j=k   followed    by  a  quadrantal  version   round 


2  2  v  2 


V 


244 

Prob.  49.  In  the  example  on  p.  459  let  b  =  250  and  c  =  500;  cal- 
culate out  the  cosine  and  the  directed  sine  of  the  product  angle. 

Prob.  50.  In  the  above  example  calculate  the  cosine  and  the 
directed  sine  up  to  and  inclusive  of  the  fourth  power  of  the  bino- 
mial. (Ans.  cos  =  .9735.) 

Prob.   51.  Calculate    the  first   four   terms   of   the   series  when 

b  =  -h,c  =  Tfa,fi  =  07/3  y  =  9^7/V!- 

Prob.  52.  From  the  fundamental  theorem  of  spherical  trigo- 
nometry deduce  the  polar  theorem  with  respect  to  both  the  cosine 
and  the  directed  sine. 

Prob.  53.  Prove  that  if  aa,  fih,  yc  denote  the  three  versors  of  a 
spherical  triangle,  then 

sin  fiy  _  sin  ya        sin  afi 


sin  a 


sin  b 


sin  c 


Art.  10.    Composition  of  Rotations. 

A  version  refers  to  the  change  of  direction  of  a  line,  but  a 
rotation  refers  to  a  rigid  body.     The  composi-     B 
tion  of  rotations  is  a  different  matter  from  the 
composition  of  versions. 

Effect  of  a  Finite  Rotation  on  a  Line. — Sup- 
pose that  a  rigid  body  rotates  6  radians  round 
the  axis  (3  passing  through  the  point  O,  and  that 
R  is  the  radius-vector  from  O  to  some  particle. 
In  the  diagram  OB  represents  the  axis  /3,  and 
OP  the  vector  R.  Draw  OK  and  OL,  the  rectangular  compo- 
nents of  R. 

fi*R  =  (cos  0  -f  sin  6  .  jf'%)rp 


46.  VECTOR    ANALYSIS    AND    QUATERNIONS.  [Chap.  IX. 

=  r(cos  0  +  sin  0  .  f?/2)(cos  fip./3  +  sin  fip . ]TpP) 
=  rfcos  fip.  /?-{-cos  0  sin  ftp .  ftpfi  -j-  sin  0  sin  fip.fip]. 
When  cos  /3p  =  o,  this  reduces  to 

/3eR  =  cos  OR  +  sin  OV(fiR). 
The  general  result  may  be  written 

0eR  =  SJ3R  .  /?  +  cos  0{V(3R){3  -f  sin  0V/S& 
Note  that  {V/3R)/3  is  equal  to  V(Vj3R)/3  because  S/l£/3  is 
o,  for  it  involves  two  coincident  directions. 

Example. — Let  /?  =  li  -f-  ?%/ -f-  «/£,  where  /2  -j-  w2  -\-?f  =  i 
and  i?  =  ;rz  +jjy'+  ^  ;  then  S/?i?  =  /*•-[-  my  -f-  #£ 
V(^)/5  = 


and 


Hence  /?*/?  =  (/;r  -j-  my  -f-  «^)(/^  +  mj  Jr  #£) 

-j-  cos  0 


mz  —  ny 

nx  —  Iz 

ly  —  mx 

I 

m               n 

i 

j               k 

V/3R  = 

I    m     n 
x  y      z 
i   j     k 

• 

mz  — 

ny 

I 

I 

i 

I 

m 

n 

x 

y 

z 

i 

j 

k 

nx  —  Iz 

ly  —  mx 

m 

n 

j 

k 

+  sin0 


To  prove  that  ftp  coincides  with  the  axis  of  yff-*/*^/"/?*/*. 
Take  the  more  general  versor  p0.  Let  OP  represent  the  axis 
/?,  AB  the  versor  /3~^2,  BC  the  versor  p0. 
Then  (AB)(BC)  =  AC  =  DA,  therefore 
>d  (AB)(BC)(AE)  =  (DA)(AE)  =  DE.  Now 
DE  has  the  same  angle  as  BC,  but  its  axis 
has  been  rotated  round  P  by  the  angle  b. 
Hence  if  0  =  tt/2,  the  axis  of  p-*/*  p«/* /&/* 
will  coincide  with  fibp* 

The  exponential   expression    for 

*  This  theorem  was  discovered  by  Cayley.  It  indicates  that  quaternion 
multiplication  in  the  most  general  sense  has  its  physical  meaning  in  the  compo- 
sition of  rotations. 


COMPOSITION    OF    ROTATIONS.  47 

fl-Wp'/'/SV*  is  e-mv/'i+\^/2+mn/\  which  may  be  expanded 
according  to  the  exponential  theorem,  the  successive  powers 
of  the  trinomial  being  formed  according  to  the  multinomial 
theorem,  the  order  of  the  factors  being  preserved. 

Composition  of  Finite  Rotations  round  Axes  which  Inter- 
sect.— Let  /3  and  y  denote  the  two  axes  in  space  round  which 
the  successive  rotations  take  place,  and  let  f3b  denote  the  first 
and  yc  the  second.  Let  fib  X  yc  denote  the  single  rotation 
which  is  equivalent  to  the  two  given  rotations  applied  in 
succession  ;  the  sign  X  is  introduced  to  distinguish  from  the 
product  of  versors.  It  has  been  shown  in  the  preceding  para- 
graph that 

ftp  =  p-*/*pr/*pb/* ; 

and  as  the  result  is  a  line,  the  same  principle  applies  to  the 
subsequent  rotation.     Hence 

yc(/3bp)  =  y-c/z(/3-b/*p"/*  P"/*)yc/z 
=  {y-</*p-b/*)p"/\l3b/*yc/2) , 
because  the  factors  in  a  product  of  versors  can  be  associated  in 
any  manner.     Hence,  reasoning  backwards, 

/3bXyc  =  (/5*V/2)2- 
Let  m  denote  the  cosine  of  /3b/'2yc/2,  namely, 

cos  b/2  cos  c/2  —  sin  b/2  sin  c/2, 
and  n.  v  their  directed  sine,  namely, 

cos  b/2  sin  c/2.y-\-zos  c/2  sin  b/2  .  /3— sin  b/2  sin  c/2  sin  0y  .  fiy\ 
then  ftb  X  yc  =  m~  —  1?  -j-  2mn .  v. 

Observation. — The  expression  (fi^y'"/*)2  is  not,  as  might  be 
supposed,  identical  with  fibyc.  The  former  reduces  to  the  lat- 
ter only  when  ft  and  y  are  the  same  or 
opposite.  In  the  figure  /3*  is  represented 
by  PQ,  yc  by  QR,  ffyc  by  PR,  /3*/»y</»  by 
ST,  and  (j3*/*ye/*)*  by  SU,  which  is  twice  p' 
ST.  The  cosine  of  SU  differs  from  the 
cosine  of  PR  by  the  term  —(sin  b/2  sin  c/2  sin  fiyf.  It  is 
evident  from  the  figure  that  their  axes  are  also  different. 


48  VECTOR    ANALYSIS    AND   QUATERNIONS. 

Corollary. — When  b  and  c  are  infinitesimals,  cos  yS*Xxc=:I> 
and  Sin  f3b  X  yc  —  b .  />  -\-  c .  y,  which  is  the  parallelogram  rule 
for  the  composition  of  infinitesimal  rotations. 

Prob.  54.  Let  /?  =  3^7/45°,  #  =  n/z,  and  JS  =  2/  -  3/  +  4^  J 
calculate  /3  i?. 

Prob.  55.  Let  /?  =  ^07/90°,  0=  7t/A,  R  =  -  /+  ay-  3*  ; 
calculate  /?  7?. 

Prob.  56.   Prove  by  multiplying  out  that  fi~  b/zp^^b/a  =  {fibp}*/3; 

Prob.  57.  Prove  by  means  of  the  exponential  theorem  that 
y-cfibyc  has  an  angle  b,  and  that  its  axis  is  yicfi. 

Prob.  58.   Prove  that  the  cosine   of   (Pb/*ye/*y  differs  from  the 

cosine  of  fihyc  by  —  (sin  -  sin  —  sin  fiy\  . 

Prob.  59.  Compare  the  axes  of  (/3b/*yc/2y  and  fibyc. 

Prob.  60.  Find  the  value  of  fib  X  yc  when   f3  =~oy/9o°  and 

y  =  po0//90°- 

Prob.  61.  Find  the  single  rotation -equivalent  to  i*/*  Xjn/Z  X  k*/K 
Prob,  62.  Prove  that  successive  rotations  about  radii  to  two 
corners  of  a  spherical  triangle  and  through  angles  double  of  those 
of  the  triangle  are  equivalent  to  a  single  rotation  about  the  radius 
to  the  third  corner,  and  through  an  angle  double  of  the  external 
angle  of  the  triangle. 


INDEX. 


Algebra  of  the  plane,  7;  of  space,  7. 
Algebraic  imaginary,  22  (footnote). 
Argand  method,  24  (footnote). 
Association  of  three  vectors,  18. 

Bibliography,  8  and  preface. 
Binomial  theorem  in  spherical  analysis, 
44. 

Cartesian  analysis,  8. 

Cayley,  46. 

Central  axis,  38. 

Coaxial  quaternions,  21;  addition  of, 
23;    product  of,  23;    quotient  of,  24. 

Complete  product  of  two  vectors,  14,  28; 
of  three  vectors,  31. 

Components  of  versor,  21;  of  quater- 
nion, 22;  of  reciprocal  of  qua- 
ternion, 22. 

Composition  of  two  simultaneous  com- 
ponents', 10;  of  any  number  of  simul- 
taneous components,  12;  of  suc- 
cessive components,  13;  of  coaxial 
quaternions,  23;  of  simultaneous 
vectors  in  space,  25;  of  mass-vec- 
tors, 35;  of  located  vectors,  36; 
of  finite  rotations,  45. 

Coplanar  vectors,  14. 

Couple  of  forces,  36;  condition  for 
couple  vanishing,  39. 

Cyclical  and  natural  order,  20. 

Determinant  for  vector  product  of 
two  vectors,  28;  for  second  par- 
tial product  of  three  vectors,  32 
and  ^y,  for  scalar  product  of  three 
vectors,  33. 


Distributive  rule,  30. 
Dynamo  rule,  27. 

Electric  motor  rule,  27. 

Exponential  theorem  in  spherical  trigo- 
nometry, 42;  Hamilton's  view,  44 
(footnote). 

Finite  rotations,  45;  versor  expression 
for,  46;  exponential  expression  for, 
46. 

Formal  laws,  12  (footnote). 

Fundamental  rules,  12  (footnote)  and  14 
(footnote) . 

Hamilton's  analysis  of  vector,  9; 
idea  of  quaternion,  21  (footnote);  view 
of  exponential  theorem  in  spherical 
analysis,  44  (footnote). 

Hay  ward,  24  (footnote). 

Hospitalier  system,  9. 

Imaginary  algebraic,  22  (footnote). 
Kennelly's  notation,  9. 
Located  vectors,  36. 

Mass-vector,  35;    composition  of,  35. 

Maxwell,  35. 

Meaning  of  dot,  9;  of  Z ,  9;  of  S,  15; 
of  V,  16;  of  vinculum  over  two  axes 
16;  of  7,  25;  of  i~  as  index,  39. 

Notation  for  vector,  9. 
Natural  order,  20. 

Opposite  vector,  17. 


50 


INDEX. 


Parallelogram  of  simultaneous  com- 
ponents, 10. 

Partial  products,  14  and  28;  of  three 
vectors,  32;  resolution  of  second 
partial  product,  33. 

Polygon  of  simultaneous  components, 
12. 

Product,  complete,  14  and  28;  par- 
tial, 14  and  28;  of  two  coplanar 
vectors,  14;  scalar,  15  and  28;  vec- 
tor, 16  and  29;  of  three  coplanar 
vectors,  18;  of  coaxial  quaternions, 
24;  of  two  vectors  in  space,  26; 
of  two  sums  of  simultaneous  vec- 
tors, 30;  of  three  vectors,  31;  of 
two  quadrantal  versors,  40;  of  two 
spherical  versors,  40;  of  three 
spherical  versors,  41. 

"Quadrantal  versor,  17. 

Quaternion,  definition  of,  21;  etymol- 
ogy of,  21  (footnote);  coaxial,  21; 
reciprocal  of,  22. 

Quaternions,  definition  of,  7;  relation 
to  vector  analysis,  7. 

Quotient  of  coaxial  quaternions,  24. 

Rayleigh,  23. 

Reciprocal  of  a  vector,  17;  of  a  qua- 
ternion, 22. 

Relation  of  right-handed  screw,  27. 

Resolution  of  a  vector,  n;  of  second 
partial  product  of  three  vectors,  33. 

Rotations,  finite,  45. 

Rules  for  vectors,  14  and  26;  for 
versors,  39;  for  expansion  of  product 
of  two  quadrantal  versors,  40;  for 
dynamo,  27. 

Scalar  product,    15;     of  two   coplanar 
vectors,  15;  geometrical  meaning,  15. 
Screw,  relation  of  right-handed,  27. 


Simultaneous  components,  9;  com- 
position of,  10;  resolution  of,  n; 
parallelogram  of,  10;  polygon  of,  12; 
product  of  two  sums  of,  30. 

Space-analysis,  7;  advantage  over 
Cartesian  analysis,  8;  foundation  of, 
12  (footnote). 

Spherical  trigonometry,  39;  funda- 
mental theorem  of,  40;  exponential 
theorem,  42;    binomial  theorem,  44. 

Spherical  versor,  40;  product  of  two, 
40;  quotient  of  two,  41;  product 
of  three,  41. 

Square  of  a  vector,  14;  of  two  simul- 
taneous components,  19;  of  two 
successive  components,  19;  of  three 
successive  components,  19. 

Stringham,  24  (footnote). 

Successive  components,  9;  composition 
of,  13. 

Tait's  analysis  of  vector,  9. 
Tensor,  definition  of,  9. 
Torque,  37. 

Total  vector  product  of  three  vectors, 
34- 

Unit-vector,  9. 

Vector,  definition  of,  8;  dimensions  of, 
9;  notation  for,  9;  unit-vector,  9; 
simultaneous,  9;  successive,  9;  co- 
planar, 14;  reciprocal  of,  17;  oppo- 
site of,  17;   in  space,  25. 

Vector  analysis,  definition  of,  7;  rela- 
tion to  Quaternions,  7. 

Vector  product,  16;  of  two  vectors,  16; 
of  three  vectors,  34. 

Versor,  components  of,  21  and  40; 
rules  for,  39;  product  of  two  quad- 
rantal, 40;  product  of  two  general 
spherical,  40;  of  three  general 
spherical,  41. 


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Compound  Riveted  Girders  as  Applied  in  Buildings 8vo,  2  00 

Planning  and  Construction  of  High  Office  Buildings 8vo,  3  50 

Skeleton  Construction  in  Buildings 8vo,  3  00 

Brigg's  Modern  American  School  Buildings 8vo.  4  00 

Carpenter's  Heating  and  Ventilating  of  Buildings 8vo,  4  00 

Freitag's  Architectural  Engineering 8vo»  3  50 

Fireproofing  of  Steel  Buildings 8vo,  2  50 

French  and  Ives's  Stereotomy.  .  , 8vo,  2  50 

1 


Gerhard's  Guide  to  Sanitary  House-inspection i6mo,    1  oo 

Theatre  Fires  and  Panics ' nmo,     i  50 

♦Greene's  Structural  Mechanics 8vo,    2  50 

Holly's  Carpenters'  and  Joiners'  Handbook i8mo,         75 

Johnson's  Statics  by  Algebraic  and  Graphic  Methods 8vo,    2  00 

Kidder's  Architects' and  Builders' Pocket-book.  Rewritten  Edition.  i6mo,mor.,  5  00 

Merrill's  Stones  for  Building  and  Decoration , 8vo,    5  00 

Non-metallic  Minerals:    Their  Occurrence  and  Uses .8vo,    4  00 

Monckton's  Stair-building 4to,    4  00 

Patton's  Practical  Treatise  on  Foundations 8vo,    5  oc 

Peabody's  Naval  Architecture 8vo,    7  50 

Richey's  Handbook  for  Superintendents  of  Construction i6mo,  mor.,    4  00 

Sabin's  industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,    3  00 

Sicbert  and  Biggin's  Modern  Stone-cutting  and  Masonry 8vo,     1  50 

Snow's  Principal  Species  of  Wood 8vo,    3  50 

Sondericker's  Graphic  Statics  with  Applications  to  Trusses,  Beams,  and  Arches. 

8vo,    2  00 

Towne's  Locks  and  Builders'  Hardware i8mo,  morocco,    3  00 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,    6  00 

Sheep,    6  50 
Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture     8vo,    5  00 

Sheep,    5  50 

Law  of  Contracts 8vo,    3  00 

Wood's  Rustless  Coatings:   Corrosion  and  Electrolysis  of  Iron  and  Steel.  .8vo,    4  00 
Worcester  and  Atkinson's  Small  Hospitals,  Establishment  and  Maintenance, 
Suggestions  for  Hospital  Architecture,  with  Plans  for  a  Small  Hospital. 

i2mo,     1  25 
The  World's  Columbian  Exposition  of  1893 Large  4to,    1  00 


ARMY  AND  NAVY. 

Bernadou's  Smokeless  Powder,  Nitro-ceilulose,  and  the  Theory  of  the  Cellulose 

Molecule 1 2mo,  2  50 

*  Bruff's  Text-book  Ordnance  and  Gunnery 8vo,  6  00 

Chase's  Screw  Propellers  and  Marine  Propulsion 8vo,  3  00 

Cloke's  Gunner's  Examiner 8vo,  1  50 

Craig's  Azimuth 4to,  3  50 

Crehore  and  Squier's  Polarizing  Photo-chronograph 8vo.  3  00 

*  Davis's  Elements  of  Law 8vo,  2  50 

*  Treatise  on  the  Military  Law  of  United  States 8vo,  7  00 

Sheep,  7  50 

De  Brack's  Cavalry  Outposts  Duties.     (Carr.) 24mo,  morocco,  2  00 

Dietz's  Soldier's  First  Aid  Handbook i6mo,  morocco,  1  25 

*  Dredge's  Modern  French  Artillery 4to,  half  morocco,  15  00 

Durand's  Resistance  and  Propulsion  of  Ships 8vo,  5  00 

*  Dyer's  Handbook  of  Light  Artillery i2mo,  3  00 

Eissler's  Modern  High  Explosives 8vo,  4  00 

•*  Fiebcger's  Text-book  on  Field  Fortification Small  8vo,  2  00 

Hamilton's  The  Gunner's  Catechism i8mo,  1  00 

*  Hoff 's  Elementary  Naval  Tactics 8vo,  1  50 

tngalls's  Handbook  of  Problems  in  Direct  Fire 8vo,  4  00 

*  Ballistic  Tables 8vo,  1  50 

*  Lyons's  Treatise  on  Electromagnetic  Phenomena.  Vols.  I.  and  II.  .8vo,  each,  6  00 

*  Mahan's  Permanent  Fortifications.    (Mercur.) 8vo,  half  morocco,  7  50 

Manual  for  Courts-martial i6mo,  morocco,  1  50 

*  Mercur's  Attack  of  Fortified  Places i2mo.  2  00 

*  Elements  of  the  Art  of  War 8vo,  4  00 

3 


Metcalf's  Cost  of  Manufactures — And  the  Administration  of  Workshops.  .8vo,  5  00 

*  Ordnance  and  Gunnery.     2  vols i2mo,  5  00 

Murray's  Infantry  Drill  Regulations i8mo,  paper,  10 

Nixon's  Adjutants'  Manual 24mo,  1  00 

Peabody's  Naval  Architecture 8vo,  7  50 

*  Phelps's  Practical  Marine  Surveying 8vo,  2  50 

Powell's  Army  Officer's  Examiner i2mo,  4  00 

Skarpe's  Art  of  Subsisting  Armies  in  War i8mo,  morocco,  1  50 

*  Walke's  Lectures  on  Explosives 8vo,  4  00 

*  .""heeler's  Siege  Operations  and  Military  Mining 8vo,  2  00 

Winthrop's  Abridgment  of  Military  Law i2mo,  2  50 

Wcodhull's  Notes  on  Military  Hygiene i6mo,  1  50 

Young's  Simple  Elements  of  Navigation i6nio,  morocco-  3  00 


ASSAYING. 

Fletcher's  Practical  Instructions  in  Quantitative  Assaying  with  the  Blowpipe. 

i2mo,  morocco,  1  50 

Furman's  Manual  of  Practical  Assaying 8vo,  3  00 

Lodge's  Notes  on  Assaying  and  Metallurgical  Laboratory  Experiments.  .  .  .8vo,  3  00 

Low's  Technical  Methods  of  Ore  Analysis 8vo,  3  00 

Miller's  Manual  of  Assaying i2mo,  1  00 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.     (Waldo.) i2mo,  2  50 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  00 

Ricketts  and  Miller's  Notes  on  Assaying 8vo,  3  00 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo, 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  00 

Wilson's  Cyanide  Processes i2mo,  1  50 

Chlorination  Process nmo,  1  50 


ASTRONOMY. 

Comstock's  Field  Astronomy  for  Engineers 8vo,  2  50 

Craig's  Azimuth 4to,  3  30 

Doolittle's  Treatise  on  Practical  Astronomy 8vo,  4  00 

Gore's  Elements  of  Geodesy 8vo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy 8vo,  3  00 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy 8vo,  2  50 

*  Michie  and  Harlow's  Practical  Astronomy 8vo,  3  00 

*  White's  Elements  of  Theoretical  and  Descriptive  Astronomy i2mo,  2  00 


BOTANY. 

Davenport's  Statistical  Methods,  with  Special  Reference  to  Biological  Variation. 

i6mo,  morocco,     1  25 

Thome  and  Bennett's  Structural  and  Physiological  Botany i6mo,    2  25 

Westermaier's  Compendium  of  General  Botany.     (Schneider.) 8vo,    2  00 


CHEMISTRY. 

Adriance's  Laboratory  Calculations  and  Specific  Gravity  Tables i2mo,  1  25 

Allen's  Tables  for  Iron'Analysis 8vo,  3  00 

Arnold's  Compendium  of  Chemistry.     (Mandel.) Small  8vo,  3  50 

Austen's  Notes  for  Chemical  Students i2mo,  1  50 

Bernadou's  Smokeless  Powder. — Nitro-cellulose,  and  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

*  Browning's  Introduction  to  the  Rarer  Elements 8vo,  1  50 

3 


Brush  and  Penfield's  Manual  of  Determinative  Mineralogy 8vo,    4  00 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.    (Bolt-wood.)-  .8vo,    3  00 

Cohn's  Indicators  and  Test-papers i2mo,    2  00 

Tests  and  Reagents 8vo,    3  00 

Craits's  Short  Course  in  Qualitative  Chemical  Analysis.   (Schaeffer.).  .  .nmo,    1  50 
Dolezalek's  Theory  of  the   Lead  Accumulator   (Storage   Battery).         (Von 

Ende.) nmo,    2  50 

Drechsel's  Chemical  Reactions.     (Merrill.) nmo,     1  25 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) 8vo,    4  00 

Eissler's  Modern  High  Explosives 8vo,    4  00 

Effront's  Enzymes  and  their  Applications.     (Prescott.) 8vo,    3  00 

Erdmann's  Introduction  to  Chemical  Preparations.     (Dunlap.) i2mo,     1  25 

Fletcher's  Practical  Instructions  in  Quantitative  Assaying  with  the  Blowpipe. 

i2mo,  morocco,     1  50 

Fowler's  Sewage  "Works  Analyses nmo,    2  00 

Fresenius's  Manual  of  Qualitative  Chemical  Analysis.     (Wells.) 8vo,    5  00 

Manual  of  Qualitative  Chemical  Analysis.  Part  I.  Descriptive.  (Wells.)  8vo,    3  00 
System   of    Instruction    in    Quantitative    Chemical   Analysis.      (Cohn.) 

2  vols 8vo,  12  50 

Fuertes's  "Water  and  Public  Health nmo,     1  50 

Furman's  Manual  of  Practical  Assaying 8vo,    3  00 

*  Getman's  Exercises  in  Physical  Chemistry nmo,    2  00 

Gill's  Gas  and  Fuel  Analysis  for  Engineers nmo,     1  23 

Grotenfelt's  Principles  of  Modern  Dairy  Practice.     (Woll.) nmo,    2  00 

Hammarsten's  Text-book  of  Physiological  Chemistry.     (Mandel.) 8vo,    4  00 

Helm's  Principles  of  Mathematical  Chemistry.     (Morgan.) nmo,     1  50 

Hering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,    2  50 

Hind's  Inorganic  Chemistry 8vo,    3  00 

*  Laboratory  Manual  for  Students nmo,     1  00 

Holleman's  Text-book  of  Inorganic  Chemistry.     (Cooper.) 8vo,    2  50 

Text-book  of  Organic  Chemistry.     ("Walker  and  Mott.) 8vo,    2  S» 

*  Laboratory  Manual  of  Organic  Chemistry.     (Walker.) nmo,     1  00 

Hopkins's  Oil-chemists'  Handbook 8vo,    3  00 

Jackson's  Directions  for  Laboratory  Work  in  Physiological  Chemistry.  .8vo,     1  25 

Keep's  Cast  Iron 8vo,    2  50 

Ladd's  Manual  of  Quantitative  Chemical  Analysis nmo,     1  00 

Landauer's  Spectrum  Analysis.     (Tingle.) 8vo,    3  00 

*  Langworthy  and  Austen.        The   Occurrence   of  Aluminium  in  Vegetable 

Products,  Animal  Products,  and  Natural  Waters 8vo,    2  00 

Lassar-Cohn's  Practical  Urinary  Analysis.     (Lorenz.) nmo,     1  00 

4pplication  of  Some   General  Reactions   to   Investigations  in  Organic 

Chemistry.     (Tingle.) nmo,    1  00 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo, 

Lob's  Electrochemistry  of  Organic  Compounds.     (Lorenz.) 8vo, 

Lodge's  Notes  on  Assaying  and  Metallurgical  Laboratory  Experiments.  ..  .8vo, 

Low's  Technical  Method  of  Ore  Analysis 8vo, 

Lunge's  Techno-chemical  Analysis.     (Cohn.) , nmo, 

Mandel's  Handbook  for  Bio-chemical  Laboratory nmo, 

*  Martin's  Laboratory  Guide  to  Qualitative  Analysis  with  the  Blowpipe.  .  nmo, 
Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

3d  Edition,  Rewritten 8vo, 

Examination  of  Water.     (Chemical  and  Bacteriological.) nmo, 

Matthew's  The  Textile  Fibres. 8vo, 

Meyer's  Determination  of  Radicles  in  Carbon  Compounds.     (Tingle.),  .nmo, 

Miller's  Manual  of  Assaying nmo, 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.     (Waldo.) .  .  .  .  nmo, 

Mixter's  Elementary  Text-book  of  Chemistry nmo, 

Morgan's  Elements  of  Physical  Chemistry nmo, 

*  Physical  Chemistry  for  Electrical  Engineers nmo, 

4 


7 

50 

3 

00 

3 

00 

3 

00 

1 

00 

1 

50 

60 

4 

00 

1 

25 

3 

50 

1 

00 

1 

00 

2 

50 

1 

50 

Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,  i  50 

Mulliken's  General  Method  for  the  Identification  of  Pure  Organic  Compounds. 

Vol.  I Large  8vo,  5  00 

O'Brine's  Laboratory  Guide  in  Chemical  Analysis 8vo,  2  00 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  00 

Ostwald's  Conversations  on  Chemistry.     Part  One.     (Ramsey.) i2mo,  1   50 

Part  Two.     (Turnbull.) nmo,  2  00 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo,  paper,  50 

Pictet's  The  Alkaloids  and  their  Chemical  Constitution.     (Biddle.) 8vo,  5  00 

Pinner's  Introduction  to  Organic  Chemistry.     (Austen.) nmo,  1  50 

Poole's  Calorific  Power  of  Fuels 8vo,  3  00 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis nmo,  1  25 

*  Reisig's  Guide  to  Piece-dyeing 8vo,  25  00 

Richards  and   Woodman's   Air,  Water,  and    Food   from  a  Sanitary  Stand- 
point  8vo,  2  00 

Richards's  Cost  of  Living  as  Modified  by  Sanitary  Science i2mo,  1  00 

Cost  of  Food,  a  Study  in  Dietaries i2mo,  1  00 

*  Richards  and  Williams's  The  Dietary  Computer 8vo,  I  50 

Ricketts  and  Russell's  Skeleton  Notes  upon  Inorganic  Chemistry.     (Part  I. 

Non-metallic  Elements.) 8vo,  morocco,  75 

Ricketts  and  Miller's  Notes  on  Assaying 8vo,  3  00 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage 8vo,  3  50 

Disinfection  and  the  Preservation  of  Food 8vo,  4  00 

Rigg's  Elementary  Manual  for  the  Chemical  Laboratory 8vo,  1  25 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo, 

Rostoski's  Serum  Diagnosis.     (Bolduan.) i2mo,  1  00 

Ruddiman's  Incompatibilities  in  Prescriptions 8vo,  2  00 

*  Whys  in  Pharmacy i2mo,  1  00 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  00 

Salkowski's  Physiological  and  Pathological  Chemistry.     (Orndorff.) 8vo,  2  50 

Schimpf's  Text-book  of  Volumetric  Analysis nmo,  2  50 

Essentials  of  Volumetric  Analysis nmo,  1  25 

*  Qualitative  Chemical  Analysis 8vo,  1  25 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses.  .  :  .  .  i6mo,  morocco,  3  00 

Handbook  for  Cane  Sugar  Manufacturers i6mo,  morocco,  3  00 

Stockbridge's  Rocks  and  Soils .8vo,  2  50 

*  Tillman's  Elementary  Lessons  in  Heat 8vo,  1  50 

*  Descriptive  General  Chemistry 8vo,  3  00 

Treadwell's  Qualitative  Analysis.     (Hall.) 8vo,  3  00 

Quantitative  Analysis.     (Hall.) 8vo,  4  00 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  00 

Van  Deventer's  Physical  Chemistry  for  Beginners.     (Boltwood.) nmo,  1  50 

*  Walke's  Lectures  on  Explosives 8vo,  4  00 

Ware's  Beet-sugar  Manufacture  and  Refining Small  8vo,  cloth,  4  00 

Washington's  Manual  of  the  Chemical  Analysis  of  Rocks 8vo,  2  00 

Wassermann's  Immune  Sera:  Haemolysins,  Cytotoxins,  and  Precipitins.    (Bol- 
duan.)   nmo,  1  00 

Well's  Laboratory  Guide  in  Qualitative  Chemical  Analysis 8vo,  1  50 

Short  Course  in  Inorganic  Qualitative  Chemical  Analysis  for  Engineering 

Students nmo,  1  50 

Text-book  of  Chemical  Arithmetic nmo,  1  25 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Wilson's  Cyanide  Processes nmo,  1  50 

Chlorination  Process nmo,  1  50 

Winton's  Microscopy  of  Vegetable  Foods 8vo,  7  50 

Wulling's    Elementary    Course    in  Inorganic,  Pharmaceutical,  and  Medical 

Chemistry nmo,  2  00 

5 


CIVIL  ENGINEERING. 

BRIDGES    AND    ROOFS.       HYDRAULICS.       MATERIALS   OF    ENGINEERING. 
RAILWAY  ENGINEERING. 

Baker's  Engineers'  Surveying  Instruments i2mo,  3  00 

Bixby's  Graphical  Computing  Table Paper  19^X24!  inches.  25 

**  Burr's  Ancient  and  Modern  Engineering  and  the  Isthmian  Canal.     (Postage, 

27  cents  additional.) 8vo,  3  50 

Comstock's  Field  Astronomy  for  Engineers 8vo,  2  50 

Davis's  Elevation  and  Stadia  Tables 8vo,  1  00 

Elliott's  Engineering  for  Land  Drainage nmo,  1  50 

Practical  Farm  Drainage nmo,  1  00 

*Fiebeger's  Treatise  on  Civil  Engineering 8vo,  5  00 

Folwell's  Sewerage.     (Designing  and  Maintenance.) 8vo,  3  00 

Freitag's  Architectural  Engineering.     2d  Edition,  Rewritten 8vo,  3  50 

French  and  I/es's  Stereotomy 8vo,  2  50 

Goodhue's  Municipal  Improvements i2mo,  1  75 

Goodrich's  Economic  Disposal  of  Towns'  Refuse 8vo,  3  50 

Gore's  Elements  of  Geodesy 8vo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy 8vo,  3  00 

Hering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Howe's  Retaining  Walls  for  Earth nmo,  1  25 

Johnson's  (J.  B.)  Theory  and  Practice  of  Surveying Small  8vo,  4  00 

Johnson's  (L.  J.)  Static's  by  Algebraic  and  Graphic  Methods 8vo,  2  00 

Laplace's  Philosophical  Essay  on  Probabilities.     (Truscoit  and  Emory.) .  nmo,  2  00 

Mahan's  Treatise  on  Civil  Engineering.     (1873.)     (Wood.).  , .  . 8vo,  5  00 

*  Descriptive  Geometry 8vo,  1  50 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy ^vo,  2  50 

Merriman  and  Brooks's  Handbook  for  Surveyors i6mo,  more-  :  -  00 

Nugent's  Plane  Surveying 8vo,  3  ~,c 

Ogden's  Sewer  Design nmo,  2  00 

Patton's  Treatise  on  Civil  Engineering 8vo  half  leather,  7  50 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  00 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage 8vo,  3  50 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry 8vo,  1  50 

Smith's  Manual  of  Topographical  Drawing.     (McMillan.) 8vo,  2  50 

Sondericker's  Graphic  Statics,  with  Applications  to  Trusses,  Beams,  and  Arches. 

8vo,  2  00 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  00 

*  Trautwine's  Civil  Engineer's  Pocket-book i6mo,  morocco,  5  00 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,  6  00 

Sheep,  6  50 
Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo,  s  00 

Sheep,  5  50 

Law  of  Contracts 8vo,  3  00 

Warren's  Stereotomy — Problems  in  Stone-cutting 8vo,  2  50 

Webb's  Problems  in  the  Use  and  Adjustment  of  Engineering  Instruments. 

i6mo,  morocco,  1  25 

Wilson's  Topographic  Surveying 8vo,  3  50 


BRIDGES  AND  ROOFS. 

Boiler's  Practical  Treatise  on  the  Construction  of  Iron  Highway  Bridges.  .8vo,  2  00 

*       Thames  River  Bridge 4to,  paper,  5  00 

Burr's  Course  on  the  Stresses  in  Bridges  and  Roof  Trusses,  Arched  Ribs,  and 

Suspension  Bridges 8vo,  3  50 


3 

00 

5 

oo 

10 

oo 

5 

oo 

3 

50 

i 

25 

2 

50 

2 

50 

2 

5<> 

2 

50 

2 

SO 

2 

50 

10 

00 

2 

00 

I 

25 

3 

S«> 

Burr  and  Falk's  Influence  Lines  for  Bridge  and  Roof  Computations.  .  .  .8vo, 

Design  and  Construction  of  Metallic  Bridges 8vo, 

Du  Bois's  Mechanics  of  Engineering.     Vol.  II Small  410, 

Foster's  Treatise  on  Wooden  Trestle  Bridges 4to, 

Fowler's  Ordinary  Foundations 8vo, 

Greene's  Roof  Trusses 8vo, 

Bridge  Trusses 8vo, 

Arches  in  Wood,  Iron,  and  Stone 8vo, 

Howe's  Treatise  on  Arches 8vo, 

Design  of  Simple  Roof-trusses  in  Wood  and  Steel 8vo, 

Johnson,  Bryan,  and  Turneaure's  Theory  and  Practice  in  the  Designing  of 

Modern  Framed  Structures Small  4to, 

Merriman  and  Jacoby's  Text-book  on  Roofs  and  Bridges: 

Part  I.     Stresses  in  Simple  Trusses 8vo, 

Part  II.     Graphic  Statics 8vo, 

Part  III.     Bridge  Design 8vo, 

Part  IV.     Higher  Structures 8vo, 

Morison's  Memphis  Bridge 4to, 

Waddell's  De  Pontibus,  a  Pocket-book  for  Bridge  Engineers.  .  i6mo,  morocco, 

Specifications  for  Steel  Bridges i2mo, 

Wright's  Designing  of  Draw-spans.     Two  parts  in  one  volume 8vo, 


HYDRAULICS. 

Bazin's  Experiments  upon  the  Contraction  of  the  Liquid  Vein  Issuing  from 

an  Orifice.     (Trautwine.) 8vo,  2  00 

Bovey's  Treatise  on  Hydraulics 8vo,  5  oo 

Church's  Mechanics  of  Engineering .  .8vo,  6  00 

Diagrams  of  Mean  Velocity  of  Water  in  Open  Channels paper,  1  50 

Hydraulic  Motors 8vo,  2  00 

Coffin's  Graphical  Solution  of  Hydraulic  Problems i6mo,  morocco,  2  501 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  00 

Folwell's  Water-supply  Engineering 8vo,  4  00 

Frizell's  Water-power 8vo,  5  00 

Fuertes's  Water  and  Public  Health i2mo,  1  50 

Water-filtration  Works i2mo,  2  50 

Ganguillet  and  Kutter's  General  Formula  for  the  Uniform  Flow  of  Water  in 

Rivers  and  Other  Channels.     (Hering  and  Trautwine.) 8vo,  4  00 

Hazen's  Filtration  of  Public  Water-supply 8vo,  3  00 

Hazlehurst's  Towers  and  Tanks  for  Water-works 8vo,  2  50 

Herschel's  115  Experiments  on  the  Carrying  Capacity  of  Large,  Riveted,  Metal 

Conduits 8vo,  2  00 

Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

8vo,  4  00 

Merriman's  Treatise  on  Hydraulics 8vo,  5  00 

*  Michie's  Elements  of  Analytical  Mechanics ; 8vo,  4  00 

Schuyler's   Reservoirs  for   Irrigation,   Water-power,   and   Domestic   Water- 
supply Large  8vo,  5  00 

**  Thomas  and  Watt's  Improvement  of  Rivers.     (Post.,  44c.  additional. ).4to,  6  00 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  00 

Wegmann's  Design  and  Construction  of  Dams 4to,  5  00 

Water-supply  of  the  City  of  New  York  from  1658  to  1895 4to,  10  00 

Williams  and  Hazen's  Hydraulic  Tables 8vo,  1  50 

Wilson's  Irrigation  Engineering Small  8vo,  4  00 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  00 

Wood's  Turbines 8vo,  2  50 

Elements  of  Analytical  Mechanics 8vo,  3  00 

7 


5 

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MATERIALS  OF  ENGINEERING. 

Baker's  Treatise  on  Masonry  Construction 8vo, 

Roads  and  Pavements 8vo, 

Black's  United  States  Public  Works Oblong  4to, 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo, 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering 8vo, 

Byrne's  Highway  Construction 8vo, 

Inspection  of  the  Materials  and  Workmanship  Employed  in  Construction. 

i6mo, 

Church's  Mechanics  of  Engineering 8vo, 

Du  Bois's  Mechanics  of  Engineering.     Vol.  I Small  4to, 

*Eckel's  Cements,  Limes,  and  Plasters 8vo, 

Johnson's  Materials  of  Construction Large  8vo, 

Fowler's  Ordinary  Foundations 8vo, 

*  Greene's  Structural  Mechanics 8vo, 

Keep's  Cast  Iron 8vo, 

Lanza's  Applied  Mechanics 8vo, 

Marten's  Handbook  on  Testing  Materials.     (Henning.)     i  vols 8vo, 

Maurer's  Technical  Mechanics 8vo, 

Merrill's  Stones  for  Building  and  Decoration 8vo, 

Merriman's  Mechanics  of  Materials 8vo, 

Strength  of  Materials i2mo, 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo, 

Patron's  Practical  Treatise  on  Foundations 8vo, 

Richardson's  Modern  Asphalt  Pavements 8vo, 

Richey's  Handbook  for  Superintendents  of  Construction i6mo,  mor., 

Rockwell's  Roads  and  Pavements  in  France i2mo,  i  25 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  00 

Smith's  Materials  of  Machines i2mo,  1  00 

Snow's  Principal  Species  of  Wood 8vo,  3  50 

Spalding's  Hydraulic  Cement i2mo,  2  00 

Text-book  on  Roads  and  Pavements nmo,  2  00 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  00 

Thurston's  Materials  of  Engineering.     3  Parts 8vo,  8  00 

Part  I.     Non-metallic  Materials  of  Engineering  and  Metallurgy 8vo,  2  00 

Part  II.     Iron  and  Steel 8vo,  3  50 

Part  IH.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Thurston's  Text-book  of  the  Materials  of  Construction 8vo,  5  00 

Tillson's  Street  Pavements  and  Paving  Materials 8vo,  4  00 

Waddell's  De  Pontibus.    (A  Pocket-book  for  Bridge  Engineers.).  .i6mo,  mor.,  2  00 

Specifications  for  Steel  Bridges i2mo,  1  25 

Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials,  and  an  Appendix  on 

the  Preservation  of  Timber 8vo,  2  00 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,  3  00 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

Steel 8vo,  4  00 


RAILWAY  ENGINEERING. 

Andrew's  Handbook  for  Street  Railway  Engineers 3x5  inches,  morocco,  1  25 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  00 

Brook's  Handbook  of  Street  Railroad  Location i6mo,  morocco,  1  50 

Butt's  Civil  Engineer's  Field-book i6mo,  morocco,  2  50 

Crandall's  Transition  Curve i6mo,  morocco,  1  50 

Railway  and  Other  Earthwork  Tables 8vo,  1  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  .  i6mo,  morocco,  5  00 


Dredge's  History  of  the  Pennsylvania  Railroad:   (1879) .  .Paper,  5  00 

*  Drinker's  Tunnelling,  Explosive  Compounds,  and  Rock  Drills. 4to,  half  mor.,  25  00 

Fisher's  Table  of  Cubic  Yards Cardboard,  25 

Godwin's  Railroad  Engineers'  Field-book  and  Explorers'  Guide.  .  .  i6mo,  mor.,  2  50 

Howard's  Transition  Curve  Field-book i6mo,  morocco,  1  50 

Hudson's  Tables  for  Calculating  the  Cubic  Contents  of  Excavations  and  Em- 
bankments  8vo,  1  00 

Molitor  and  Beard's  Manual  for  Resident  Engineers i6mo,  1  00 

Nagle's  Field  Manual  for  Railroad  Engineers i6mo,  morocco,  3  00 

Philbrick's  Field  Manual  for  Engineers i6mo,  morocco,  3  00 

Searles's  Field  Engineering i6mo,  morocco,  3  00 

Railroad  Spiral i6mo,  morocco,  1  50 

Taylor's  Prismoidal  Formula?  and  Earthwork 8vo,  1  50 

*  Trautwine's  Method  of  Calculating  the  Cube  Contents  of  Excavations  and 

Embankments  by  the  Aid  of  Diagrams 8vo,  2  00 

The  Field  Practice  of  Laying  Out  Circular  Curves  for  Railroads. 

i2mo,  morocco,  2  50 

Cross-section  Sheet Paper,  25 

Webb's  Railroad  Construction i6mo,  morocco,  5  00 

Wellington's  Economic  Theory  of  the  Location  of  Railways Small  8vo,  5  00 


DRAWING. 

Barr's  Kinematics  of  Machinery 8vo,    2  50 

*  Bartlett's  Mechanical  Drawing 8vo,    3  00 

*  "                    "                    "        Abridged  Ed 8vo,  1  50 

Coolidge's  Manual  of  Drawing 8vo,  paper  1  00 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  Engi- 
neers  Oblong  4to,  2  50 

Durley's  Kinematics  of  Machines 8vo,    4  00 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo,    2  50 

Hill's  Text-book  on  Shades  and  Shadows,  and  Perspective 8vo,    2  00 

Jamison's  Elements  of  Mechanical  Drawing 8vo,    2  50 

Advanced  Mechanical  Drawing 8vo,     2  00 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery 8vo,     1  50 

Part  n.     Form,  Strength,  and  Proportions  of  Parts 8vo, 

MacCord's  Elements  of  Descriptive  Geometry 8vo, 

Kinematics;  or,  Practical  Mechanism 8vo, 

Mechanical  Drawing 4to, 

Velocity  Diagrams 8vo, 

MacLeod's  Descriptive  Geometry Small  8vo, 

*  Mahan's  Descriptive  Geometry  and  Stone-cutting 8vo, 

Industrial  Drawing.     (Thompson.) 8vo, 

Moyer's  Descriptive  Geometry gvo,  2  00 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  00 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  00 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  00 

Robinson's  Principles  of  Mechanism 8vo,  3  00 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  co 

Smith's  (R.  S.)  Manual  of  Topographical  Drawing.     (McMillan.) 8vo,  2  50 

Smith  (A.  W.)  and  Marx's  Machine  Design 8vo,  3  00 

Warren's  Elements  of  Plane  and  Solid  Free-hand  Geometrical  Drawing.  i2mo,  1  00 

Drafting  Instruments  and  Operations i2mo,  1  25 

Manual  of  Elementary  Projection  Drawing i2mo, 

Manual  of  Elementary  Problems  in  the  Linear  Perspective  of  Form  and 

Shadow i2mo,  1  00 

Plane  Problems  in  Elementary  Geometry i2mo,  1  23 

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3 

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3 

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3 

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5 

00 

2 

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4 

00 

3 

00 

Warren's  Primary  Geometry i2mo,         75 

Elements  of  Descriptive  Geometry,  Shadows,  and  Perspective 8vo,  3  50- 

General  Problems  of  Shades  and  Shadows 8vo,  3  o& 

Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Problems,  Theorems,  and  Examples  in  Descriptive  Geometry 8vo,  2  50 

Weisbach's    Kinematics    and    Power    of    Transmission.        (Hermann    and 

Klein.) 8vo,  5  o0, 

Whelpley's  Practical  Instruction  in  the  Art  of  Letter  Engraving i2mo,  2  00 

Wilson's  (H.  M.)  Topographic  Surveying 8vo,  3  50 

Wilson's  (V.  T.)  Free-hand  Perspective 8vo,  2  50 

Wilson's  (V.  T.)  Free-hand  Lettering 8vo,  1  00 

Woolf's  Elementary  Course  in  Descriptive  Geometry Large  8vo,  3  00 

ELECTRICITY  AND  PHYSICS. 

Anthony  and  Brackett's  Text-book  of  Physics.     (Magie.) Small  8vo,    3  00 

Anthony's  Lecture-notes  on  the  Theory  of  Electrical  Measurements.  .  .  .  i2mo, 
Benjamin's  History  of  Electricity 8vo, 

Voltaic  Cell 8vo, 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.     (Boltwood.).8vo, 

Crehore  and  Squier's  Polarizing  Photo-chronograph 8vo, 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  i6mo,  morocco, 
Dolezalek's    Theory    of    the    Lead    Accumulator    (Storage    Battery).      (Von 

Ende.) i2mo, 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) 8vo, 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo, 

Gilbert's  De  Magnete.     (Mottelay.) 8vo,    2  50 

Hanchett's  Alternating  Currents  Explained i2mo,     1  00 

Hering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,    2  50 

Holman's  Precision  of  Measurements 8vo,    2  00 

Telescopic   Mirror-scale  Method,  Adjustments,  and  Tests.  ..  .Large  8vo,         75 

Kinzbrunner's  Testing  of  Continuous-current  Machines 8vo,    2  00 

Landauer's  Spectrum  Analysis.     (Tingle.) 8vo, 

Le  Chatelier's  High-temperature  Measurements.  (Boudouard — Burgess.)  i2mo, 
Lob's  Electrochemistry  of  Organic  Compounds.     (Lorenz.) 8vo, 

*  Lyons's  Treatise  on  Electromagnetic  Phenomena.   Vols.  I.  and  II.  8vo,  each, 

*  Michie's  Elements  of  Wave  Motion  Relating  to  Sound  and  Light 8vo, 

Niaudet's  Elementary  Treatise  on  Electric  Batteries.     (Fishback.) i2mo, 

*  Rosenberg's  Electrical  Engineering.     (Haldane  Gee — Kinzbrunner.).  .  .8vo, 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     Vol.  1 8vo, 

Thurston's  Stationary  Steam-engines 8vo, 

*  Tillman's  Elementary  Lessons  in  Heat 8vo, 

Tory  and  Pitcher's  Manual  of  Laboratory  Physics Small  8vo,    2  00 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,    3  00 

LAW. 

*  Davis's  Elements  of  Law 8vo,    2  50 

*  Treatise  on  the  Military  Law  of  United  States 8vo,    7  00 

*  Sheep,    7  5<> 

Manual  for  Courts-martial i6mo,  morocco,     1  50 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,    6  00 

Sheep,  6  50 
Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo  5  00 

Sheep,  5  50 

Law  of  Contracts 8vo,  3  00 

Winthrop's  Abridgment  of  Military  Law i2mo,  2  So 

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3 

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6 

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4 

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1 

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2 

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2 

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1 

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MANUFACTURES. 

Bernadou's  Smokeless  Powder — Nitro-cellulose  and  Theory  of  the  Cellulose 

Molecule nmo,  2  50 

Bolland's  Iron  Founder nmo,  2  50 

"The  Iron  Founder,"  Supplement i2mo,  2  50 

Encyclopedia  of  Founding  and  Dictionary  of  Foundry  Terms  Used  in  the 

Practice  of  Moulding i2mo,  3  00 

Eissler's  Modern  High  Explosives 8vo,  4  00 

Effront's  Enzymes  and  their  Applications.     (Prescott.) 8vo,  3  00 

Fitzgerald's  Boston  Machinist i2mo,  1  00 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  1  00 

Hopkin's  Oil-chemists'  Handbook 8vo,  3  00 

Keep's  Cast  Iron 8vo,  2  50 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control Large  8vo,  7  50 

Matthews's  The  Textile  Fibres. 8vo,  3  50 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo,  2  00 

Metcalfe's  Cost  of  Manufactures — And  the  Administration  of  Workshops. 8vo,  5  00 

Meyer's  Modern  Locomotive  Construction 4to,  10  00 

Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,  1  50 

*  Reisig's  Guide  to  Piece-dyeing 8vo,  25  00 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  00 

Smith's  Press-working  of  Metals 8vo,  3  00 

Spalding's  Hydraulic  Cement i2mo,  2  00 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco,  3  00 

Handbook  for  Cane  Sugar  Manufacturers i6mo,  morocco,  3  00 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  00 

Thurston's  Manual  of  Steam-boilers,  their  Designs,  Construction  and  Opera- 
tion  8vo,  5  00 

*  Walke's  Lectures  on  Explosives 8vo,  4  00 

Ware's  Beet-sugar  Manufacture  and  Refining Small  8vo,  4  00 

West's  American  Foundry  Practice i2mo,  2  50 

Moulder's  Text-book i2mo,  2  50 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  00 

Wood's  Rustless  Coatings :   Corrosion  and  Electrolysis  of  Iron  and  Steel.  .8vo,  4  00 


MATHEMATICS. 

Baker's  Elliptic  Functions 8vo,  I  50 

*  Bass's  Elements  of  Differential  Calculus .  nmo,  4  00 

Briggs's  Elements  of  Plane  Analytic  Geometry nmo,  1  00 

Compton's  Manual  of  Logarithmic  Computations nmo,  1  50 

Davis's  Introduction  to  the  Logic  of  Algebra 8vo,  1  50 

*  Dickson's  College  Algebra Large  i2mo,  1  50 

*  Introduction  to  the  Theory  of  Algebraic  Equations Large  nmo,  1  25 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo,  2  50 

Halsted's  Elements  of  Geometry 8vo,  1  75 

Elementary  Synthetic  Geometry 8vo,  1  50 

Rational  Geometry nmo,  1  7S 

*  Johnson's  (J.  B.)  Three-place  Logarithmic  Tables:   Vest-pocket  size. paper,  15 

100  copies  for  5  00 

*  Mounted  on  heavy  cardboard,  8X10  inches,  25 

10  copies  for  2  00 

Johnson's  (W.  W.)  Elementary  Treatise  on  Differential  Calculus .  .Small  8vo,  3  00 

Johnson's  (W.  W.)  Elementary  Treatise  on  the  Integral  Calculus. Small  8vo,  1  50 

11 


Johnson's  (W.  W.)  Curve  Tracing  in  Cartesian  Co-ordinates i2mo,     i  oo 

Johnson's  (W.  W.)  Treatise  on  Ordinary  and  Partial  Differential  Equations. 

Small  8vo,    3  50 
Johnson's  (W.  W.)  Theory  of  Errors  and  the  Method  of  Least  Squares.  12 mo,     1  50 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,    3  00 

Laplace's  Philosophical  Essay  on  Probabilities.     (Truscott  and  Emory.) .  i2mo,    2  00 

*  Ludlow  and  Bass.     Elements  of  Trigonometry  and  Logarithmic  and  Other 

Tables 8vo,    3  00 

Trigonometry  and  Tables  published  separately Each,    2  00 

*  Ludlow's  Logarithmic  and  Trigonometric  Tables 8vo,     1  00 

Mathematical  Monographs.     Edited  by  Mansfield  Merriman  and  Robert 

S.  Woodward Octavo,  each     1  00 

No.  1.  History  of  Modern  Mathematics,  by  David  Eugene  Smith. 
No.  2.  Synthetic  Projective  Geometry,  by  George  Bruce  Halsted. 
No.  3.  Determinants,  by  Laenas  Gifford  Weld.  No.  4.  Hyper- 
bolic Functions,  by  James  McMahon.  No.  5.  Harmonic  Func- 
tions, by  William  E.  Byerly.  No.  6.  Grassmann's  Space  Analysis, 
by  Edward  W.  Hyde.  No.  7.  Probability  and  Theory  of  Errors, 
by  Robert  S.  Woodward.  No.  8.  Vector  Analysis  and  Quaternions, 
by  Alexander  Macfarlane.  No.  9.  Differential  Equations,  by 
William  Woolsey  Johnson.  No.  10.  The  Solution  of  Equations, 
by]  Mansfield  Merriman.  No.  1 1.  Functions  of  a  Complex  Variable, 
by  Thomas  S.  Fiske. 

Maurer's  Technical  Mechanics 8vo,    4  00 

Merriman  and  Woodward's  Higher  Mathematics 8vo,    5  00 

Merriman's  Method  of  Least  Squares 8vo,    2  00 

Rice  and  Johnson's  Elementary  Treatise  on  the  Differential  Calculus. .  Sm.  8vo,    3  00 

Differential  and  Integral  Calculus.     2  vols,  in  one Small  8vo,    2  50 

Wood's  Elements  of  Co-ordinate  Geometry 8vo,    2  00 

Trigonometry:  Analytical,  Plane,  and  Spherical i2mo,    1  00 


MECHANICAL  ENGINEERING. 
MATERIALS  OF  ENGINEERING,  STEAM-ENGINES  AND  BOILERS. 

Bacon's  Forge  Practice i2mo,  1  50 

Baldwin's  Steam  Heating  for  Buildings i2mo,  2  50 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

*  Bartlett's  Mechanical  Drawing 8vo,  3  00 

*  "  "  "        Abridged  Ed 8vo,  1  50 

Benjamin's  Wrinkles  and  Recipes i2mo,  2  00 

Carpenter's  Experimental  Engineering 8vo,  6  00 

Heating  and  Ventilating  Buildings 8vo,  4  00 

Cary's  Smoke  Suppression  in  Plants  using  Bituminous  Coal.     (In  Prepara- 
tion.) 

Clerk's  Gas  and  Oil  Engine Small  8vo,  4  00 

Coolidge's  Manual  of  Drawing 8vo,  paper,  1  00 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  En- 
gineers  Oblong  4to,  2  50 

Cromwell's  Treatise  on  Toothed  Gearing i2mo,  1  50 

Treatise  on  Belts  and  Pulleys i2mo,  1  50 

Durley's  Kinematics  of  Machines 8vo,  4  00 

Flather's  Dynamometers  and  the  Measurement  of  Power i2mo,  3  00 

Rope  Driving i2mo,  2  00 

Gill's  Gas  and  Fuel  Analysis  for  Engineers i2mo,  1  25 

Hall's  Car  Lubrication i2mo,  1  00 

Hering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

12 


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3 

00 

5 

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2 

00 

4 

00 

4 

00 

5 

oo 

4 

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50 

i 

50 

3 

50 

3 

00 

Hutton's  The  Gas  Engine 8vo, 

Jamison's  Mechanical  Drawing 8vo, 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery 8vo, 

Part  H.     Form,  Strength,  and  Proportions  of  Parts 8vo, 

Kent's  Mechanical  Engineers'  Pocket-book i6mo,  morocco, 

Kerr's  Power  and  Power  Transmission 8vo, 

Leonard's  Machine  Shop,  Tools,  and  Methods 8vo, 

*  Lorenz's  Modern  Refrigerating  Machinery.    (Pope,  Haven,  and  Dean.)  .  .  8vo, 
MacCord's  Kinematics;   or,  Practical  Mechanism 8vo, 

Mechanical  Drawing 4to, 

Velocity  Diagrams ; 8vo, 

MacFarland's  Standard  Reduction  Factors  for  Gases 8vo, 

Mahan's  Industrial  Drawing.     (Thompson.) 8vo, 

Poole's  Calorific  Power  of  Fuels 8vo, 

Reid's  Course  in  Mechanical  Drawing 8vo,    2  00 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,    3  00 

Richard's  Compressed  Air i2mo,     1  50 

Robinson's  Principles  of  Mechanism 8vo,    3  00 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,    3  00 

Smith's  (O.)  Press-working  of  Metals 8vo,    3  00 

Smith  (A.  W.)  and  Marx's  Machine  Design 8vo,    3  00 

Thurston's   Treatise   on   Friction  and   Lost   Work   in   Machinery   and   Mill 

Work 8vo,    3  00 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics.  i2mo,     1  00 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,    7  so 

Weisbach's    Kinematics    and    the    Power    of    Transmission.     (Herrmann — 

Klein.) 8vo,    5  00 

Machinery  of  Transmission  and  Governors.     (Herrmann — Klein.).  .8vo,    5  00 

Wolff's  Windmill  as  a  Prime  Mover 8vo,    3  00 

Wood's  Turbines 8vo,    2  50 


MATERIALS  OP  ENGINEERING. 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering.    6th  Edition. 

Reset 8vo,  7  50 

Church's  Mechanics  of  Engineering 8vo,  6  00 

*  Greene's  Structural  Mechanics 8vo,  2  50 

Johnson's  Materials  of  Construction 8vo,  6  00 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Martens's  Handbook  on  Testing  Materials.     (Henning.) 8vo,  7  50 

Maurer's  Technical  Mechanics 8vo,  4  00 

Merriman's  Mechanics  of  Materials 8vo,  5  00 

Strength  of  Materials i2mo,  1  00 

Metcalf's  Steel.     A  manual  for  Steel-users i2mo,  2  00 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  00 

Smith's  Materials  of  Machines nmo,  1  00 

Thurston's  Materials  of  Engineering 3  vols.,  8vo,  8  00 

Part  II.     Iron  and  Steel 8vo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Text-book  of  the  Materials  of  Construction 8vo,  5  00 

Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials  and  an  Appendix  on 

the  Preservation  of  Timber 8vo,  2  00 

13 


Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,  3  00 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

Steel 8vo,  4  00 

STEAM-ENGINES  AND  BOILERS. 

Berry's  Temperature-entropy  Diagram nmo,  1  25 

Carnot's  Reflections  on  the  Motive  Power  of  Heat.     (Thurston.) nmo,  I  5a 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  .  .   i6mo,mor.,  5  00 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  1  00 

Goss's  Locomotive  Sparks 8vo,  2  00 

Hemenway's  Indicator  Practice  and  Steam-engine  Economy nmo,  2  00 

Hutton's  Mechanical  Engineering  of  Power  Plants 8vo,  5  00 

Heat  and  Heat-engines 8vo,  5  00 

Kent's  Steam  boiler  Economy 8vo,  4  00 

Kneass's  Practice  and  Theory  of  the  Injector 8vo,  1  5a 

MacCord's  Slide-valves 8vo,  2  oa 

Meyer's  Modern  Locomotive  Construction 4to,  10  00 

Peabody's  Manual  of  the  Steam-engine  Indicator i2mo.  1  50 

Tables  of  the  Properties  of  Saturated  Steam  and  Other  Vapors    8vo,  1  00 

Thermodynamics  of  the  Steam-engine  and  Other  Heat-engines 8vo,  5  00 

Valve-gears  for  Steam-engines 8vo,  2  50 

Peabody  and  Miller's  Steam-boilers 8vo,  4  00 

Pray's  Twenty  Years  with  the  Indicator Large  8vo,  2  50 

Pupin's  Thermodynamics  of  Reversible  Cycles  in  Gases  and  Saturated  Vapors. 

(Osterberg.) i2mo,  1  25 

Reagan's  Locomotives:   Simple   Compound,  and  Electric i2mo,  2  50 

Rontgen's  Principles  of  Thermodynamics.     (Du  Bois.) 8vo,  5  00 

Sinclair's  Locomotive  Engine  Running  and  Management nmo,  2  00- 

Smart's  Handbook  of  Engineering  Laboratory  Practice nmo,  2  50 

Snow's  Steam-boiler  Practice 8vo,  3  00 

Spangler's  Valve-gears 8vo,  2  50 

Notes  on  Thermodynamics nmo,  1  00 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo,  3  00 

Thurston's  Handy  Tables 8vo,  1  50 

Manual  of  the  Steam-engine 2  vols.,  8vo,  10  00 

Part  I.     History,  Structure,  and  Theory 8vo,  6  00 

Part  II.     Design,  Construction,  and  Operation 8vo,  6  00 

Handbook  of  Engine  and  Boiler  Trials,  and  the  Use  of  the  Indicator  and 

the  Prony  Brake 8vo,  5  00 

Stationary  Steam-engines 8vo,  2  50 

Steam-boiler  Explosions  in  Theory  and  in  Practice nmo,  1  50 

Manual  of  Steam-boilers,  their  Designs,  Construction,  and  Operation 8vo,  5  00 

Weisbach's  Heat,  Steam,  and  Steam-engines.     (Du  Bois.) 8vo,  5  00 

Whitham's  Steam-engine  Design 8vo,  5  00 

Wilson's  Treatise  on  Steam-boilers.     (Flather.) i6mo,  2  50 

Wood's  Thermodynamics,  Heat  Motors,  and  Refrigerating  Machines.  .  .8vo,  4  00 


MECHANICS  AND  MACHINERY. 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures   8vo,  7  50 

Chase's  The  Art  of  Pattern-making nmo,  2  50 

Church's  Mechanics  of  Engineering 8vo,  6  00 

Notes  and  Examples  in  Mechanics 8vo,  2  00 

Compton's  First  Lessons  in  Metal-working nmo,  1  50 

Compton  and  De  Groodt's  The  Speed  Lathe 12 mo,  1  50. 

14 


Cromwell's  Treatise  on  Toothed  Gearing i2mo,  I  50 

Treatise  on  Belts  and  Pulleys i2mo,  1  50 

Dana's  Text-book  of  Elementary  Mechanics  for  Colleges  and  Schools,  .nmo,  1  50 

Dingey's  Machinery  Pattern  Making i2mo,  2  00 

Dredge's  Record  of  the  Transportation  Exhibits  Building  of  the  "World's 

Columbian  Exposition  of  1893 4to  half  morocco,  5  00 

Du  Bois's  Elementary  Principles  of  Mechanics : 

Vol.      I.     Kinematics 8vo,  3  50 

Vol.    n.     Statics 8vo,  4  00 

Mechanics  of  Engineering.     Vol.    I Small  4to,  7  50 

Vol.  H Small  4to,  10  00 

Durley's  Kinematics  of  Machines 8vo,  4  00 

Fitzgerald's  Boston  Machinist i6mo,  1  00 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  00 

Rope  Driving i2mo,  2  00 

Goss's  Locomotive  Sparks 8vo,  2  00 

*  Greene's  Structural  Mechanics 8vo,  2  50 

Hall's  Car  Lubrication i2mo,  1  00 

Holly's  Art  of  Saw  Filing i8mo,  75 

James's  Kinem~  tics  of  a  Point  and  the  Rational  Mechanics  of  a  Particle. 

Small  8vo,  2  00 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,  3  00 

Johnson's  (L.  J.)  Statics  by  Graphic  and  Algebraic  Methods 8vo,  2  00 

Jones's  Machine  Design: 

Part    I.     Kinematics  of  Machinery 8vo,  1  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  00 

Kerr's  Power  and  Power  Transmission 8vo,  2  00 

Lanza's  Applied  Mechanics 8vo,  7  50 

Leonard's  Machine  Shop,  Tools,  and  Methods 8vo,  4  00 

*  Lorenz's  Modern  Refrigerating  Machinery.     (Pope,  Haven,  and  Dean.). 8vo,  4  00 
MacCord's  Kinematics;   or,  Practical  Mechanism 8vo,  5  00 

Velocity  Diagrams 8vo,  1  50 

Maurer's  Technical  Mechanics 8vo,  4  00 

Merriman's  Mechanics  of  Materials 8vo,  5  00 

*  Elements  of  Mechanics i2mo,  1  00 

*  Michie's  Elements  of  Analytical  Mechanics 8vo,  4  00 

Reagan's  Locomotives:   Simple,  Compound,  and  Electric nmo,  2  50 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  00 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  00 

Richards's  Compressed  Air i2mo,  1  50 

Robinson's  Principles  of  Mechanism 8vo.  3  00 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     Vol.  1 8vo,  2  50 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  00 

Sinclair's  Locomotive-engine  Running  and  Management i2mo,  2  00 

Smith's  (0.)  Press-working  of  Metals 8vo,  3  00 

Smith's  (A.  W.)  Materials  of  Machines i2mo,  1  00 

Smith  (A.  W.)  and  Marx's  Machine  Design 8vo,  3  00 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo,  3  00 

Thurston's  Treatise  on  Friction  and  Lost  Work  in    Machinery  and    Mill 

Work 8vo,  3  00 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics. 

i2mo,  1  00 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Weisbach's Kinematics  and  Power  of  Transmission.   (Herrmann — Klein.). 8vo,  5  00 

Machinery  of  Transmission  and  Governors.      (Herrmann — Klein. ).8vo,  5  00 

Wood's  Elements  of  Analytical  Mechanics 8vo,  3  00 

Principles  of  Elementary  Mechanics i2mo,  1  25 

Turbines 8vo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  1  00 

15 


METALLURGY. 

Egleston's  Metallurgy  of  Silver,  Gold,  and  Mercury: 

Vol.    I.     Silver 8vo,  7  50 

Vol.  II.     Gold  and^Mercury 8vo,  1  50 

**  Iles's  Lead-smelting.     (Postage  0  cents  additional.) nmo,  2  50 

Keep's  Cast  Iron 8vo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo,  1  50 

Le  Chatelier's  High-temperature  Measurements.  (Boudouard — Burgess. )i2mo,  3  00 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo,  2  00 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.     (Waldo.). . .  .  i2mo,  2  50 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo, 

Smith's  Materials  of  Machines i2mo,  1  00 

Thurston's  Materials  of  Engineering.     In  Three  Parts 8vo,  8  00 

Part    II.     Iron  and  Steel 8vo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  00 


MINERALOGY. 


Barringer's  Description  of  Minerals  of  Commercial  Value.    Oblong,  morocco,  2  50 

Boyd's  Resources  of  Southwest  Virginia 8vo,  3  00 

Map  of  Southwest  Virignia Pocket-book  form.  2  00 

Brush's  Manual  of  Determinative  Mineralogy.     (Penfield.) 8vo,  4  00 

Chester's  Catalogue  of  Minerals 8vo,  paper,  1  00 

Cloth,  1  25 

Dictionary  of  the  Names  of  Minerals 8vo,  3  50 

Dana's  System  of  Mineralogy Large  8vo,  half  leather,  12  50 

First  Appendix  to  Dana's  New  "  System  of  Mineralogy." Large  8vo,  1  00 

Text-book  of  Mineralogy 8vo,  4  00 

Minerals  and  How  to  Study  Them i2mo,  1  50 

Catalogue  of  American  Localities  of  Minerals Large  8vo,  1  00 

Manual  of  Mineralogy  and  Petrography nmo,  2  00 

Douglas's  Untechnical  Addresses  on  Technical  Subjects i2mo,  1  00 

Eakle's  Mineral  Tables 8vo,  1  25 

Egleston's  Catalogue  of  Minerals  and  Synonyms 8vo,  2  S» 

Hussak's  The  Determination  of  Rock-forming  Minerals.    ( Smith.). Small  8vo,  2  00 

Merrill's  Non-metallic  Minerals:  Their  Occurrence  and  Uses 8vo,  4  00 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo,  paper,  50 
Rosenbusch's   Microscopical   Physiography   of   the   Rock-making  Minerals. 

(Iddings.) 8vo,  5  00 

*  Tillman's  Text-book  of  Important  Minerals  and  Rocks 8vo,  2  00 


MINING. 

Beard's  Ventilation  of  Mines i2mo,  2  50 

Boyd's  Resources  of  Southwest  Virginia 8vo,  3  00 

Map  of  Southwest  Virginia Pocket-book  form  2  00 

Douglas's  Untechnical  Addresses  on  Technical  Subjects nmo,  1  00 

*  Drinker's  Tunneling,  Explosive  Compounds,  and  Rock  Drills.  .4to,hf.  mor.,  25  00 

Eissler's  Modern  High  Explosives 8vo,  4  00 

16 


Fowler's  Sewage  Works  Analyses i2mo,  2  00 

Goodyear's  Coal-mines  of  the  Western  Coast  of  the  United  States nmo,  2  50 

Ihlseng's  Manual  of  Mining 8vo,  5  00 

**  lles's  Lead-smelting.     (Postage  ox.  additional.) i2mo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo,  1  50 

O'Driscoll's  Notes  on  the  "treatment  of  Gold  Ores 8vo,  2  00 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo, 

*  Walke's  Lectures  on  Explosives 8vo,  4  00 

Wilson's  Cyanide  Processes i2mo,  1  50 

Chlorination  Process i2mo,  1  50 

Hydraulic  and  Placer  Mining i2mo,  2  00 

Treatise  on  Practical  and  Theoretical  Mine  Ventilation i2mo,  1  25 


SANITARY  SCIENCE. 

Bashore's  Sanitation  of  a  Country  House i2mo, 

Folwell's  Sewerage.     (Designing,  Construction,  and  Maintenance.) 8vo, 

Water-supply  Engineering 8vo, 

Fuertes's  Water  and  Public  Health i2mo, 

Water-filtration  Works i2mo, 

Gerhard's  Guide  to  Sanitary  House-inspection i6mo, 

Goodrich's  Economic  Disposal  of  Town's  Refuse Demy  8vo, 

Hazen's  Filtration  of  Public  Water-supplies 8vo, 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo, 

Mason's  Water-supply.  (Considered  principally  from  a  Sanitary  Standpoint)  8vo, 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2mo, 

Ogden's  Sewer  Design i2mo, 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis i2mo, 

*  Price's  Handbook  on  Sanitation i2mo, 

Richards's  Cost  of  Food.     A  Study  in  bietaries i2mo,     1  00 

Cost  of  Living  as  Modified  by  Sanitary  Science i2mo,  1  00 

Richards  and  Woodman's  Air.  Water,  and  Food  from  a  Sanitary  Stand- 
point  8vo,  2  00 

*  Richards  and  Williams's  The  Dietary  Computer 8vo,  1  50 

Rideal's  Sewage  and  Bacterial  Purification  of  Sewage 8vo,  3  50 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  00 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo,  1  00 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Winton's  Microscopy  of  Vegetable  Foods 8vo,  7  50 

Woodhull's  Notes  on  Military  Hygiene i6mo ,  1  50 


MISCELLANEOUS.  . 

De  Fursac's  Manual  of  Psychiatry.     (Rosanoff  and  Collins.).  . .  .Large  i2mo,  2  50 
Emmons's  Geological  Guide-book  of  the  Rocky  Mountain  Excursion  of  the 

International  Congress  of  Geologists Large  8vo,  1  50 

Ferrel's  Popular  Treatise  on  the  Winds 8vo.  4  00 

Haines's  American  Railway  Management i2mo,  2  50 

Mott's  Fallacy  of  the  Present  Theory  of  Sound  i6mo,  1  00 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute,  1824-1894. . Small  8vo,  3  00 

Rostoski's.Serum  Diagnosis.     (Bolduan.) i2mo,  1  00 

Rotherham's  Emphasized  New  Testament Large  8vo,  2  00 

17 


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50 

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4 

00 

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Steel's  Treatise  on  the  Diseases  of  the  Dog 8vo.  3  50 

The  World's  Columbian  Exposition  of  1893 4to,  1  00 

Von  Behring's  Suppression  ot  Tuberculosis.     (Bolduan.) i2mo,  1  00 

Winslow's  Elements  of  Applied  Microscopy nmo,  I  SO 

Worcester  and  Atkinson.     Small  Hospitals,  Establishment  and  Maintenance; 

Suggestions  for  Hospital  Architecture :  Plans  for  Small  Hospital .  i2mo,  1  25 


HEBREW  AND  CHALDEE  TEXT-BOOKS. 

Green's  Elementary  Hebrew  Grammar nmo,  1  25 

Hebrew  Chrestomathy 8vo,  2  00 

Gesenius's  Hebrew  and  Chaldee  Lexicon  to  the  Old  Testament  Scriptures. 

(Tregelles.) Small  4to,  half  morocco,  5  00 

Letteris's  Hebrew  Bible 8vo,  2  25 

18 


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